You can get a very good thermal scope for 2k or less, it just depends on how picky you are.
Basically the only thing higher cost thermals give you is better resolution if your goal is to hunt below 200yards.
I have a HikMicro Lynx LC06 that was around $400. Terrible resolution but it can detect animals well over 100yards but it depends on how big they are.
Downside is you can't really tell what you are looking at; I was out one night testing it, saw some cats, a hedgehog and what I thought was a frog in the grass.
I stood there for probably 10 seconds trying to figure out what type of animal it was, it was pitch black so all I had to go on was the thermal scope.
I was just about to grab it with my hand to capture it but as I wasn't sure what it was I didn't want to get bit so I brought out my phone and turned on the camera light... it was a giant poop my 200 pound Leonberger dog had just laid...
When things are moving it's easier to figure out what they are when the resolution is garbage.
When you say small/medium game are we talking rabbit sized up to deer? At 100 yards it might be hard to tell the difference between a cat and a fox with a low resolution thermal scope.
I can't really give any recommendations as high end thermals are outside my budget at the moment and don't have much experience with them. Cheap thermal for spotting and a "high end" night vision scope is one option. If it has to be thermal then try to find the highest resolution one you can find, at $2000 it will probably be 256x192 but I'd try for a 384x288 if possible. Maybe try to find a used one from someone upgrading to a better one.