Yes you're correct, the CO2 is liquefied under pressure so it just flows in, its not actually "pumped".
It does settle to the bottom of the tank (or sits on top of the wine), which displaces the lighter air mix. Because there is a venting valve at the top of the tank, the air is pushed out and eventually the entire volume is filled with CO2.

A fancy setup can actually monitor the gas pressure and automatically top it up as needed.

Probably a bit fancy for a 5gal bucket, but I have heard of people using soda siphon bulbs to inject CO2 into smaller wine vessels, but I’m not sure exactly how.