Savage or Stevens, a couple of barrels and/or stocks. You can also easily swap trigger groups if you want a light pull. All the performance you want without breaking the bank. If you stick to one bolt size, you can use any cartridge within that case head family.
The barrel swapping takes a few minutes and can easily be done at home. Just be careful checking the headspace.
if you want a d mag rifle, the new center feed mag is superb. Just waiting for the stock aftermarket to catch up. We should see tactical, hunting and varminting stocks to take advantage of this system. Really is smooth and will feed AI wildcats without hanging up. The 308 length mags are also quite long inside allowing for OAL close to 3.00".
As to cartridge, a one size fits all would likely be from the 308 family. The 308 is a no brainer. But I think the 7-08 as a better compromise. The 243 and 260 lean more towards the lighter game in my books even though there is no shortage of hunters who harvest moose size game every year with these.
However, with swapping barrels, you could have a 22/250, 22BR, or 6BR for the PD popping and go to a 338-06 or 35 Whelen or 40cal if you want a big thumper, and everything in between that will fit in a standard action mag.
If you can get different bolt heads, then any cartridge is at your disposal.
Modular in design and a lot of fun to swap things. Unfortunately, getting tough to get parts due to restrictions from the US govt and Savage.
There is little problem building a few rifles within the budget you listed.
Jerry