Yes, the PA38 has crap trim because it isn't a trim tab. It is a spring pushing against the yoke (in effect). The mechanism is guaranteed to not work properly after a while, whereas a trim tab will always work so long as it is properly lubed because it works aerodynamically.
I've drained about five beakers of water from a PA38, before any avgas started coming out - after a rainy night.
Another gripe I have with the PA38 is that it is a waste of the punter's money because they are paying for flight training in a plane which they are highly unlikely to be flying post-PPL. There are very few of them on the rental scene, and very few people choose to fly them for real even when they can rent one.
The idea of getting a PPL is to learn to fly and fly from A to B (unless you want to do aeros, etc). It is not to throw money at becoming a better pilot in the abstract.
And the hardest thing will be to do map+stopwatch nav in a less than stable plane. Why not just remove the yoke and fly with the trim and the pedals? It is perfectly possible and would make one an even better pilot
I've got about 20hrs in a PA38... never again.