Find tuned vertical movement component for arduino
A solenoid should do what you want.  Your objection of "does not offer fine tuned movement" makes no sense.  You haven't specified what "fine tuned movement" even is, nor what about it a solenoid can't do.  You also haven't given any justification for why you need this, whatever this is.

You want to press a pushbutton.  The pushbutton will either be open or closed.  It doesn't matter whether you press it a little or a lot to close it, as long as you don't push hard enough to cause damage.  The switch will be closed either way.

A solenoid does exactly that.  The travel for activating a pushbutton is usually small, maybe around 1 mm.  The minimum force is also small, and there is a wide range between hard enough and too hard.

I really don't see how a solenoid doesn't fit your criteria, other than the made up ones with no justification.