After playing around with Main Story quest "Lure the Matsugane Family Out", which is chapter 10, i came to a couple of conclusions.

To progress the story as fast as possible you need to:

1. Trigger Kaito's dialogue in the small alleyway near the Nakamichi Alley.
2. Run a certain amount of distance to get his 2nd dialogue which also progresses the story and sends you to Tender.

Here are short videos of different approaches that you can use, those didn't have any enemies in it - despawn them with the Drone if needed - especially important to not be in any fight when heading to Nakamichi Alley Kaito's dialogue.

1. With going straight inside the Bar Tender and running around in it (1m15s):

2. With going to the Tender and just doing circles near the entrance (1m4s).

3. With passing Kaito's dialogue and running through random alleys for a bit (1m2s):

Timings are from [getting control] to [activating Mari's dialogue].

The one with going straight to Bar Tender i have tested the most, but still unsure if you need to wreck chaos inside, or you can just run inside the bar without touching chairs and stuff. One thing is for certain - running into the walls doesn't help, it doesn't get counted as distance traveled fast enough and ends up taking over 3 minutes.

Apparently, you can fly through the 2nd door in the Bar Tender with your drone, which isn't useful here, unfortunately.