Best cheap integrated amp

The Yamaha AS 801 is about 899.00 I believe. The AA 501 is approximately 500.00.

Cambridge Audio has the CXA80 on Ebay quite often at $450.00,
80 watts per ch with a decent DAC.

Accessories4less has this unit which is pretty nice.

DENON PMA-800NE Stereo Integrated Amplifier w/built-in
DAC and Phono Preamplifier $419.99

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.