Why is acceleration negative in a free fall?

Technically speaking altitude is measured from the ground up.

Therefore, say you're falling from 1000 metres, that number decreases till it reaches 0. Therefore, you're accelerating but towards the ground i. e. towards 0.

Therefore acceleration is negative because you're travelling at - x metres per second and accelerating at - x metres per second^2

Only because when most people work with vertical motion, they call the upward direction positive. So falling objects have negative velocity and acceleration. But if you prefer to call the downward direction positive, you have every right to do that. Then the force of gravity on a falling object, as well as its velocity and acceleration, are all positive, and its negative altitude steadily grows until it hits the ground.