TL;DR: birds can fly up and down so they are more likely to get lost

Expanding on what OP is talking about:

In this context, a random walk happens on a 2D coordinate plane. Your drunk person starts at the origin, (0, 0), and for a "random walk" they move either left, right, up, or down by exactly 1 unit each step. It's a mathematical fact that this process, taken to its limit where infinitely many random steps are taken, will always have the drunk return to the origin - in fact, for any given integer coordinate on the plane there's a 100% chance the drunk will eventually visit that coordinate following a random walk.

This doesn't work in 3D though, where there's an x, y, and z axis. A random walk there won't always return to the origin - it only will about 34% of the time. If the drunk gets too far away the probability of ever finding their way back at random quickly drops to 0.

Gravity gives up and down a lack of symmetry that left and right maintain. A drunk bird is more likely to fall than fly to the moon.

Also, in materials science, randomly walking atoms (Brownian motion) in a crystal structure displace roughly at, if I recall, the square root of their velocity, on average.

Everyone is always equally drunk, and the road is always the same

Guys, an unpopular opinion, but perhaps you shouldn't take too long showers, eh?

I don't think a drunk bird can fly

I dont get it

But a drunk bird will start from the ground, and probably end on the ground, which collapses its random walk back to a 2d plane

The thing is that drunken walking/flying is still not completely random.