I have a PSL and an older 110 and love them both.
However,
A PSL:
you are more limited in options for ammunition as it uses a relatively light bullet.
Parts for repairs can be found but finding a smith to work on a surplus rifle can be a challenge.
Limited scope options unless you mount an adapter.
It has reloading capability but you have to find reloadable brass. Most rounds are not reloadable but are actually old surplus.
Rounds are usually not found in chain stores, but only guns shops
.308 or equivalent 30cal rifle:
An average to nice .308 caliber leaves many more options that often cost less.
most any smith will work on one
you can get a used .308 bolt action at a low enough price to allow you to customize it to your needs and stay well under your $1000 limit.
You can find a .308 in the AR platform to get a semi-auto, but it will most likely go over the $1000 mark
rounds come in MANY MANY weights and powder loads from many manufacturers you can find ammunition in just about any place that sells ammo.
reloading is a breeze and you can load it to your wants with just about any configuration
scope mounting is fairly standardized.
Tons of accessories available if you go with the AR platform
Now,
If you a looking for a rifle to just shoot, that is accurate and fun, AND semi-auto is what you want, then a PSL is a great rifle since you can get stup for less than the $1000 mark.
Just my thoughts and good luck which ever way you go.