You can absorb enemy systems ‘peacefully’ with influence in the late-game, so it can be good to produce a truck load of it! Blocking off systems you don’t want to colonize (solitary gas giants, etc.) to stop enemy colonization could be good too.
I think you need enough for your laws. That can often be more than you’d produce without help (even 1 per population is rare without specific anomalies, buildings or heroes). The stockpile is for diplomacy, but that’s not much of a factor right now, to be honest…
In peacetime, the pressure bar seems fairly pointless - if basically lets a big empire with a lot of influence make ‘demands’ of an empire with less influence once the bar fills up. If they refuse the demand, you get to go to war for ‘free’ (without the influence cost). That’s not all that great.