Best bang for buck integrated amp for $500

The Yamaha S801 is a nice integrated. I bought one for my son and it does a nice job driving his Harbeth C7's. You should be able to find a used one for a little more than your stated budget. 

Get a used Arcam FMJ A19. Kinda on the "warm" side with a great MM phono stage. 50 WPC. Used ones range from $400 to $500. Otherwise, a used Rega Brio, not a new IO. Also 50 WPC with a good MM phono stage and a small footprint. I thought the Arcam had a better soundstage, but the Brio is quite nice.

NAD 316beev2 on my desk with Elac DBR62, Rega P2/Exact2, Luxman DZ CD & T-102 tuner (with outdoor antenna), and Schitt Modi 3 connected to the PC.

The NAD is shallow (~ 9.5 inches) so it fits on the desk in spite of traditional 17" width.  Simple and sounds great with anything but speakers with too much treble emphasis, even the integrated phono stage, which is dead quiet.  I love having a button for each source rather than searching for a source by clicking or spinning a dial through.  I think the build quality is better than the prior few generation NAD entry models and the desktop models.

If you are going to buy used in this price range, I would steer away from stuff that is too old unless you can do the work yourself or like paying for and waiting for repairs.  I don't think you can get a used Rega Brio that isn't 10+ years old in your price range, but maybe you will get lucky.

Also, I had the Yamaha AS-301 and wasn't impressed with the phono stage in particular.  That range of amps is also quite large and can dominate a smaller space.