Best cheap integrated amp

Used Rega Brio-R can be had for about $500 or less.  Certainly not the last word in resolution or finesse, but a lot of fun at that price point.

As I'm sure you already know, the synergy between amp and speakers is paramount. Some amps sound great with some speakers and lousy with others. There are plenty of NAD opponents here but I can make a recommendation that's worth looking into.....
The NAD C326BEE (approx $500) is a very musical amp capable of driving speakers above what it's power rating may imply. I currently have the NAD C375BEE and first used it with Klipsch Reference speakers. Didn't like the sound. Then used the amp with PSB speakers. Night and day difference. NAD and PSB are/were sister companies.

If you have the chance to audition an integrated amp of interest, try to audition it with the speakers you intend to buy. Same with any other combination of amp and speakers.

Probably going to get flak for this but on a budget for a pre-pro so I landed on the Yamaha RX-A780. It was the only one around $500 with a set of pre-outs so I could hook up my amp to push the mains. I have to say it’s not bad and the room correction is decent. Also it allows me to bi-amp my front “presence” speakers as I don’t have rear surrounds. So, Mains, Presence, Center and Sub. The Yamaha has enough options for adjustments and is doing a decent job for the $$$.

I was thinking used on the Yamaha.

I have MusicHall a15.3 in my bedroom system and its absolutely wonderful with the right speakers (I have it hookd up to Kef Q150 - and they have a good synergy). $450 retail.. plenty of inputs. The only thing I don't like about it is the remote - tiny buttons.