For music sure but what about 60% movies? A centre channel is a must for movie watching. It does the heavy lifting of any video presentation. I've tried stereo setup with phantom centre channel but it was not even close to the real thing.
I would still go for your original advice having a stereo amp with HT by-pass. If the budget is limited why not upgrade the AVR first with pre-outs. Then when the budget allows add a stereo amp. BTW, $2k will buy you a very nice AVR (especially second hand current/recent models) with improved sound quality so you may not feel the urge to upgrade anytime soon.