What are some brands and models of receivers that I should be on the lookout for in my price range?
And if I do get a new receiver, would I be able to use the connects from the subwoofer driver to connect it to the new receiver? If that were possible, I'd just use the same enclosure I currently have with a slight modification.
Forget trying to salvage your current sub. Take it to the recycling unit and end your relationship with it.
I can't answer your question precisely, unless I have a clear idea of how you want to use your system and what you want it to do.
I suspect at the minimum you need a 5.1 receiver with at least Dolby digital and DTS. You will need a powered sub. You can keep your current satellite speakers for the moment, but keep you eyes open for something much better.
I don't know if brand matters a lot. If it is older and still working fine, that tells you most of what you want to know.
Now when I said equipment is not serviceable, I should amplify that a bit.
Generally modern equipment is only serviceable by board change and that does not need circuits and a service manual as a rule.
The reasons are that modern boards use surface mount components, that can only be placed by robots and not humans. The law only requires boards be kept available for seven years after the last production.
Boards are made once and when gone they are gone. It frequently happens boards are gone before seven years. It is hard for a manufacturer to know how many boards to produce.
The next issue is the ICs to even make the boards. A lot of the ICs will be custom. Only one production run of almost all but the most popular ICs have only one production run, because of cost of set up. When gone they are gone. That is why older equipment with tubes and discrete transistors are so much easier to service.
I have had to give up in nice older units now and again or wait a long time for what I want to come up on eBay, because a custom IC has been NLA and usually for a long time.
You are not viewing the modern world of electronics from the correct window.
I would be 99% certain your unit is not serviceable by anyone. The reason being the boards are not serviceable due to type of construction and lack of service information and the boards being NLA a long time ago.
So this is my last piece of advice: - Move on!