I've done a little bit of glitch hunting and this is the closest I've come to a skip so far. If you open the menu before you enter the slide after delivering Luke Belmont's pizza, then when you close the menu you gain the ability to stack your movement onto the movement caused by the slide, causing a very unintended displacement (the displacement in the video was caused by just walking forward through the entire slide).

If you displace out of bounds and fall into the void, there seems to be a "death plane" of sorts beneath the map that warps you to the bridge in the middle of the city when you land on it. Displacing out of bounds and then warping like this would save 17-19 seconds over doing everything as-intended, but unfortunately entering the weird slide-walking state seems to cause a softlock at the end of the game that prevents you from sliding down the pole. If you exit the game and re-open it, you can actually slide down the pole now, but that takes far longer than 17-19 seconds so that's definitely slow.

I have another theory, though. If the game auto-saves while you're on the slide, and you exit and re-enter the game, you exit the sliding state and you just fall through the slide to the ground (this is probably why resetting the game removes the softlock). You can inconsistently force the game to auto-save while you're on the slide by opening your inventory and waiting a second or two, so my theory is to use the slide to displace over Caetano's, force the game to auto-save, exit+reload the game and fall somewhere out of bounds in/around Caetano's. And then, from there, there would need to be a way to navigate to either the saxophone room or to the train.

In summary, if anyone can find a way to avoid the softlock that would save up to 17-19 seconds, and if anyone can find a way to displace directly to the end using the slide then whoa boy that saves a lot more.