A few thoughts from a layman:
What you are talking about is pressure and it doesn't come into play here as the snooker ball is very hard, solid object that moves away from the cue the moment it is hit. (Unlike you being hit by a football boot) .
The speed the ball leaves the cue depends on the speed the cue hits it. It will move just as fast/far if hit by you using a pinhead or with a mallet provided they were travelling at the same speed when they hit the ball(disregarding the differences in friction of different substances).
In niether snooker or pool is the object to hit the balls as hard as possible, it is to make the balls go where you want them to. The reason for hitting the balls with a small size head is to be able to hit the ball on the sides/top/bottom and create spin to control the ball after it has been hit, i.e. swerve, backspin, topspin, etc.
Professional pool balls are bigger than professional snooker balls hence having bigger cues.
The cues provided by the pub are thick because they are poor quality and so that people don't keep snapping them. The members of the pub pool team will have their own cues that have small tips like snooker cues.