>I just fall backwards, every time.

Without knowing anything, I'd say check your back angle - it's probably too vertical. If you're falling backwards, point your nipples at the floor.

https://youtu.be/QhVC_AnZYYM?t=427

But really, just find a weightlifting coach and spend some money to get some coaching sessions. They can fix this stuff in minutes.

And if you find the vertical back angle more natural, you should try high bar squatting. With high bar you will also benefit from having your heels slightly raised with a plate under your heel or weightlifting shoes.

Your ankle to knee and knee to hip ratio can make this easier/harder due to centre of gravity position.

The three areas I've identified to work on for myself are ankle mobility, hip flexibility and hamstring flexibility.

I realised that hamstring flexibility impairs my hip mobility, and my hamstrings are good by average standards - I can touch my knuckles on the ground from standing position after warming up.

Are you saying the more flexible your hamstrings are the less mobile your hips are or am I reading that wrong?

My hamstrings are super tight (can't barely get to my ankles when bending over) so any tips on what you did to increase your hamstring flexibility would be welcome too!

No, the opposite, if your hamstrings are tight then when you are trying to flex your hips you're also having to fight against your hamstrings at the same time.

Hamstring stretches are pretty well trodden, yoga or any kicking martial art will have a wealth of material for how to increase your hamstring flexibility.

I literally learned only a couple months ago that not being able to squat like that is a thing. (And no, I'm not a third worlder either). Incidentally I also learned a while back that I have somewhat hypermobile joints so maybe that's that.

See my comment at the same level as yours, it might help.