I decided to inquire about it in #gta. I tried it and elaborated on what I had seen Powdinet do before on stream. Turns out you can probably do Taxi Driver in around 20 minutes.

1) Bring a Zebra Cab to the Hyman Condo in Downtown. Save.

2) Save a replay over the minigun rampage in downtown (or whatever nearby rampage you prefer). I'll probably stick to this since it's the closest in location and would reduce risk of load time stuff. Also it fits the best with route changes for me anyway. Wait for four minutes, two for the insta-rampage failing, two for the duping.

3) Load save. Rampage-start taxi, then once you get onto the Trial By Dirt bike, start counting the dupes up from 2 (since this'll reflect how many fares you end up getting). Press F3, then press F1-F3 in very quick succession so that you can cancel the rampage and restart it in this tiny window before Trial By Dirt restarts itself.

This is a pretty quick way I figured you could do it. It may seem inconsistent at first, and yes it is very difficult to get consistently. It's just -really- quick succession with the F1 and F3, but not so quick that you press them at the same time and screw yourself over.

If it starts without a rampage, you're screwed. Load the game and try again.

4) Dupe until you have counted to 50 (or 49). Start taxi and drop off a fare. Return to Hyman Condo and repeat (new replay and all) but with the amount of dupes you'll need to reach 99 fares (otherwise crash [?]).

5) Deliver the final fare. At this point I'd guess you'd want to do the minigun rampage as well regardless of your route.

APPARENTLY doing 100 fares in two sessions works without crashing as well, I'll look into that.

4m (replay) + 2m (duping) + 3m (estimate of worst-case scenario taxi fare and back downtown) + 4m (replay) + 2m (duping) + 1m30 (estimate of worst-case scenario taxi fare) + 1m30 (estimate of worst-case scenario taxi fare) = 18 minutes if you don't fuck up

You could also make one continuous replay for both super-duped Trial By Dirts but that's kinda risky imo, in case you get a super shitty fare, and waiting for long enough to guarantee it to be safe is a time waste.

Video of me failing horrifically and showing that this will actually take practice:

4 minute break in hundo is probably the best thing about this.

I think you could also use the Python rampage, Minigun is closer but in my case this would require crapload of rerouting because of Vigilante. If you care about every second you should consider replay buying PW, Skumole Shack or Hyman Condo (PW works great for me, for others this may be an overally bad idea)
This sounds scary, but on the other hand, you get those 8 minutes to do stuff IRL in the middle of the run Kreygasm

I'm retarded, for some stupid reason I can't understand I thought that timing starts when you get to the rampage. I also forgot that zebra cab spawns in Kaufmann depot after Cabmaggedon, lol

pretty sure zebra isn't worth it, also any rampage in downtown/little haiti should work

The Zebra Cab is a consistent spawn and is fast in case you get shitty-ass fares, and it's not too far from the Hyman Condo anyway.

just die in a mission and you'll get a consistent taxi spawn meters away from hyman condo ultiDuping

Which you can't take because your guns are gone and pressing F takes you to somewhere else. Could maybe die during publicity tour or something though

You can complete the mission without picking up any fares. You do some bullshit setup earlier in the run, TbD start Taxi Driver and drive to the destination you had set up. Repeat. Skip to TL;DR, if you are not interested in technicalities.
Video:
When TbD starting Taxi Driver, all Taxi Driver instances but one start at a point where the game assumes there to be a fare and is in the middle of generating a destination for you. In order to do that, it generates a random number which I call destination_id and then picks the destination based on destination_id. Since TbD starting skip to the middle of this process, it assumes that there is already a destination_id and does not generate a new one. The one from the last fare is used. Due to all this, two things can happen.

1) If your last fare has given you a destination_id of 11 or higher, you get the corresponding destination - but from the first island. Bad.
2) If you never had a fare or the last fare has given you a destination_id of 10 or lower, the game does not pick a new destination and the destination from the last fare is used instead. If you never had a fare, that destination is (0 | 0 | 0) - unreachable.

TL;DR
At any point in the run, start "Taxi Driver" on the second island and pick up a fare. Do this until you get one of the following destinations, the others don't work. Then cancel Taxi Driver and the setup is done. After TbD duping, start Taxi Driver, drive to your destination, repeat. Don't let anything confuse you - like markers disappearing.

[good destinations in bold]
destination_id - destination name - coordinates
1 'Ammu-Nation' in Downtown. -679.1 1193.5 9.4
2 'the Terminal' at Escobar International Airport. -1457.7 -825.1 13.1
3 'Sunshine Autos' in Little Havana. -1009.0 -880.9 11.9
4 'Kaufman Cabs' in Little Haiti. -1018.4 195.9 10.1
5 'The Hardware store' in Little Havana. -971.2 -677.2 10.4
6 'Howlin Petes Bike Emporium' in Downtown. -600.6 667.3 9.9
7 'Police Station' in Little Havana. -865.6 -677.0 10.0
8 'Police Station' in Downtown. -661.7 755.7 9.8
9 'Hospital' in Downtown. -800.4 1167.8 9.6
10 'Hospital' in Little Havana. -871.9 -467.2 9.8

If you make a longer rampage replay, you can watch the replay until you see Tommy pick up the rampage and react accordingly.

- If you see the rampage disappear during the replay, it means the rampage start was successful. You can stop the replay with F1 or F3, and then press F3 F1 F3 to do it again.

- If you don't see the rampage appear at all it means that Tommy either successfully picked the rampage before it appeared, or that you were too quick with F3 F1 F3. If latter, you can salvage this by pressing F1 F3 quickly (as done with the first two rampage starts in the video), and if former, you can do it just to be safe, as you can't really tell.

It's still possible to fuck up by being too slow, but as far as I can tell, it makes TbD duping a lot easier, as you essentially can't be too quick with the replays.

This should let you continue duping after failure. It is slow, though and you lose a Sanchez every time you do it, so you need to stockpile a bunch of those beforehand. Maybe you get them the quickest by driving to TbD and warping back to the Hyman Condo a bunch of time, I don't know if the TbD Sanchez respawns immediately.

So you failed rampage starting TbD and have om1. Now you die - om 0, one TbD instance fails. Now you - as Gael commented on YouTube - run to Hyman Condo (I warped in the video, using a replay collecting both a ramapge and save icon), get on your Sanchez and drive back to TbD to continue duping.

Someone test this properly, please, I have to go.

There's just enough time to run to the garage from the hospital and start taxi before TbD fails, so leaving taxi in the garage should be a really nice back up strat. You can just keep duping until you fail and then roll with it.

E: Recorded another example video. Used a replay where Tommy walks into the rampage marker to avoid the situation where you don't see the rampage appear at all, still pretty fast. Used Patrick's backup strat after failing final dupe at around 1:50 mark. Properly done taxi should take under 15 minutes.

I'd personally have a Sanchez parked as well and get on and off to reset the timer for safety. ^^

Pussy strat 2.0:
In the fareless setup, you get the hospital destination. Setup done. (read the post about fareless). I get one attempt every 3s once I get in the situation of getting the same fare every other time I start taxi (you see the destination, cancel and restart taxi twice and the guy who just got out of your taxi is your new fare). Just try it out to see what I mean.

In the Taxi part
1) you record a 10min(dunno - more?)-into-the-future replay.
2) You can glitch a taxi's passenger door and it will never despawn (except for save/load).
3) You can then park it at the hospital

You cannot load without losing the taxi then, but mistakes become less costly: you set up a 10 min rampage replay. If you get om1 TbD, you just die, get into the taxi, start taxi. Since it is already at the hospital, you immediately get the fare progress and can get out of the taxi.
Then you run to the Love Fist car and drive back to TbD, picking up weapons from Hyman Condo on the way for the next death. Repeat until you run out of time or get the 100 fares.

Small update on stuff that was found ages ago, but never bothered to post here:

It was found that rampage starting TbD is affected by saving and loading the game. More specifically, it affects the order in which the mission scripts are loaded. So if you have the bad order, TbD can't be rampage started consistently, and you have save and load in order to fix it. However, if you have the right script order, duping TbD is consistent as long as you don't fuck up. What it means is that during runs you just need to count the amount times you save and load - even numbers=good, odd=bad.

It was also found that it can also be used to instapass Vigilante, you just have to dupe TbD 11 times and then start Vigilante. This can crash the game, but, if you save and restart the whole game before doing it, it's consistent.

Optimally the way you'd do the whole ordeal is by using Patrick's method to force the taxi fare destination to Downtown. In which case, you can do both rounds of taxi duping and vigilante in one go with just one replay (you'll still be able to save the game between rounds of duping). I've yet to figure out the exact wait timing for it.

[anti's posts restoration project]
It is worth noting that completing TbD normally before Taxi + Vigilante duping is dangerous, because starting it again would create objects (barrels). Duping this over and over becomes very unstable and can cause crashes.