You push in the centre first because the cap makes a seal when you put it on. Pushing the centre in when the cap is already sealed compresses the air in the tank.
The point of the centre of the cap is to compensate for expansion of air in the tank when you put in chemicals warmer than the air in the tank (warmer air occupies greater volume than colder air) or pour in a chemical that, when it contacts what's already in the tank, creates a gas of some sort (I think bleach-fix does that when it contacts residual developer on film in the tank). The expanding gas in the tank pushes the centre of the cap out instead of just popping the cap off - and then pouring out any chemical that is inside, if the volume is enough to fill past the halfway mark.
When developing colour film, when the chemicals are fresh, I will need to stand by and wait for the centre to pop out so I can quickly stand the tank up, unseal the cap, push the centre back in, reseal the cap, and resume. Otherwise, the cap does pop off from the pressure.