Edit: I have rewritten this post to be more concise and have all information in one post. (up to EBen's May 2018 post)
Demo by anti:

I barely if at all cover micro-optimization except when it makes the instapass more consistent (like getting rid of the ambulance passenger door to start Paramedic in time). For those you need to watch a run. Powdinet and Ractrot have posted some info for optimization/routing, the rest is covered here.

Why you should safety save (before this instapass)
1. The game might crash when you approach the office. It does not happen often, though, but it has happened in runs. (Tech: The mission fail marks several models as no longer needed. So they get unloaded by some part of the game, eventually, (but I don't know when) causing a game crash when you reach the office.)
2. You might softlock/crash by making a mistake.

Prerequisites
- copcar
- call or clothes replay (to gain control during a cutsene)

Steps
1. Start Vigilante with OM = 0 (using the save point).
2. Start G-spotlight.
3. Fail it once by getting busted.
- After getting busted, you absolutely have to respawn on the Western island. [note1]
- Failing G-spotlight again crashes the game (dying with om0 does not crash, though).
4. Suicide for another 12h advance to avoid 7:00
- the time being exactly 7:00 fails the mission.
- You can avoid saving by some precise replays to skip 7:00 (replays add 1 to the in-game minute) https://pastebin.com/0ixPkbMt
5. Acquire both an ambulance and any bike, park them near the elevator.
- Do not park too close to the elevator marker to avoid it despawning:
¤ Park it outside the office.
- Also, get rid of the passenger door (!) since the timing will be tight later.
6. Hold a call, get on PCJ and trigger elevator cutscene.
7. Let go of call (or use a clothes replay), get off the bike, quickly enter the Ambulance and trigger Paramedic.
- Just hold A/Sprint and press F when you are near the Ambulance.
- You have 2 seconds after the cutscene starts. If you have practise, park properly, and get rid of the passenger door then that is safe.
8. Leave the Ambulance to cancel the other Paramedic instance.

If you fail to gain control in the lift cutscene...
... then you cannot instapass anymore. You can salvage the situation, though:
- If you trigger the lift cutscene again and do the same thing you tried to do the first time(start Paramedic in that cs), a different wait offset is used than the first time. It does not instapass but it safely gets rid of G-spotlight, so that you can try again (starting from the very beginning: get a cop vehicle).
- If you leave your bike but fail to get into the Ambulance in time then you most likely crash:
- If you start Paramedic, the game most likely crashes (untested).
- If you do not, the mission goes on but you softlock after leaving the lift. The clock reaching 7:00 should (!) fail the mission, crashing the game.
- You can use replays to see where Tommy is, maybe you can start a fight and get killed, if that helps...

optimizations
1) I proposed grinding against that wall to remove the door, anti:"tried that once and flipped so never again".

2) Powdinet on the optimal suicide place: "you can do it before the load trigger, just slightly before the lamppost"

Update: According to anti, this description does not apply to the busted warp, but "almost never going to get a cop car to chase you that early" so that it is not often needed to know where exactly you need to be.

More info
1) To know where you are going inside the office you can use either
- the first person camera(probably it is the way the cinematic camera works in-doors)
- or look back. Warning: holding "look back" during the "get-out" animation (after pressing F) starts Paramedic, crashing the game.
- or look to the side

Notes
[note1]
(about respawning at the wrong hospital)
Else you will most likely crash the game when you are near the office. So just load your safety save. If you do not have a save then you are not allowed to drive past the Film Studios without the PCJ model being loaded. A PCJ replay unloads the PCJ model, btw, the opposite of the hoped effect. [I have heard opposite "opinions" about replays unloading models]

Make sure the cop car does not have a passenger, or else.
https://clips.twitch.tv/TenderRelentlessZucchiniDancingBaby

Credits
mainly go to husho for finding the call stack shenanigans. I then found the IP using that, and the 10 (in total) test runs were done by both of us.

I did not want to just delete the old post. ^^ Here it is, whatever.
https://pastebin.com/yHg3sth7

with this new big strat sub 6 is not a thing anymore i think

GTA III 1st Powdinet 5:48:24
That's the thing now. ^^ Albeit an easier target.

Edit: From reading the scm/hex, on the japanese version, it should also work. It executes one "wait" (opcode 9020) less, otherwise it works the same.

Edit2:
Vigilante can be used instead of Bombs Away. Vigilante cannot die in this game, it seems from reading the scm. Also, we have this one peace of evidence that it cannot die: It does not die from using it for the instapass.

From what I can tell, this is the method:

1. Start Bombs Away (or Vigilante? why use this instead if it can't "die")
2. Force OM = 0 (rampage completion)
3. Start G-Spotlight
4. Force OM = 0 again (die)
5. Save to avoid 7:00, because that time fails the mission
6. Acquire both an ambulance and a PCJ, park them near the elevator used in G-Spotlight
7. Hold a call, get on PCJ and trigger elevator cutscene
8. Let go of call, get off PCJ, enter ambulance and trigger Paramedic
9. Instapass!

Am I correct?

It also seems (presently) not worth the crash possibility until we figure out what exactly causes models to unload.

Yes, pretty much.

Vigilante should be faster overall, as long as you route in a good Police Car. My idea is the following:

1. Get a FBI Rancher from the end of Martha's Mugshot (alternatively, can fly the heli to the Vice Point or Downtown police station, or even do this part at a later point in the run, for example, after TBD Vigilante)
2. Savewarp start Vigilante, start G-Spotlight
3. Proceed with the instapass

The model unloading crash has never happened to me and from what I've read neither has Patrick or husho experienced one, but even so, you can save the game for safety, and this instapass saves so much time that even with 2-3 crashes it would still save time.

I had crashes back when I drove from the other hospital (or teleported from the other hospital). With this hospital, husho and me never crashed in 10 attempts using Bombs Away (+3 extra by myself with Vigilante and +unknown by Powdinet with unknown mission), unless there were more tests where one crashed. ^^

"Vigilante never dies" means that you can still start Vigilante after starting a mission during it - like taxi in old taxi duping or G-spotlight. That's ("die") how people phrase it when talking about this. ^^

I edited your list:
1. Start Vigilante with OM = 0 (using the save point)
2. Start G-spotlight
3. Fail G-spotlight once by dying
4. Save to avoid 7:00(exactly 7:00), because that time fails the mission
6. Acquire both an ambulance and any bike, park them near the elevator used in G-spotlight
7. Hold a call, get on PCJ and trigger elevator cutscene
8. Let go of call, get off the bike, quickly enter ambulance and trigger Paramedic
9. Instapass!

about 8: just hold A/Sprint and press F when you are near the Ambulance. Also, get rid of the passenger door.

Also, failing G-spotlight a second time crashes the game.

I think people should still read the whole thread since it contains some good-to-know info not in the list. ^^

Fantastic information and discovery, thank you to everyone involved 🙂

Would probably be faster to save at Skumole Shack. With a good PCJ spawn on the road it should be possible to pull it off without saving, but when's that going to happen? 😛

Tested vigi method on JP, works.

I have rewritten the intro post to have all information in one post.

E: @ EBen's post below me: fixed

Realised the video on this thread died a while back, so here's a non dead one