Gotcha, I always assumed you HAD to find the source of that bell ringing considering how much they focused on it. Good to know! The whole chasing Yuan thing is prolly my least favorite part of the game honestly so anything to skip some of it could help lmfao... I will try this soon on my latest playthrough...
That doesn't skip any of following Yuan; you arrive at the roof of Three Birds Bldg. in all cases, and following her is required.
If you don't mind using glitches, you can skip lots of following Yuan without skipping anything else, and it doesn't matter how you find out about the bird shop (orphanage, knife shop, or direct):
When following Yuan and she leaves the Tea Break Bldg. heading for Thousand White Bldg., as soon as you get outside Yuan goes to the left; head right, through the Stand Qr., and straight into Moon Child Bldg. Congratulations, you've just skipped three big cutscenes and three "let's keep following her" cutscenes, as well as a ton of walking.
Not enough? Before Yuan catches the elevator down in Tea Break Bldg., use the map glitch to get to the elevator without triggering the cutscene. That's another one down.
Still not enough? Fine. Instead of going into Moon Child Bldg., go into Black Heaven Bldg. instead. Head up to the 5F, and then run down the first half of the steps. Line yourself up with the second half, and do the map glitch on the very top step of the staircase, exiting out when you're in Moon Child Bldg. Then just run around the corner and you've finished following Yuan. This brings the following Yuan section from 5 large cutcsenes, 8 small cutscenes, and 9 walking sections to 0 large, 3 small, and 2 walking sections.
Alternatively, do what I do and buy capsule toys and drinks while following her.