good integrated amps that don't include a phono section or a DAC

Primaluna. Classic versions, Prologue, Dialogue, roll tubes to fine tune sound. Plus  tubes sound great with digital (and of course any other sources). New or used. Simplicity, what is needed - nothing more. 

Cambridge Audio Azur 851a

120WPC, no DAC, no Phono.  Neutral, and slightly warm sound.

Under $1k used most of the time.

I think Rotel is a great band for the buck company but people say they can be pretty bright with the wrong speakers. I never thought so.

I the used market I can vouch for Primare, Creek (especially the CLASSIC 5350SE),and Belles Aria. I also loved a Prima Luna tube integated I heard in a showroom; but I didn't investigate the speakers it was driving as I was not there to shop that day.

I chose a used Primare I32 class D integrated myself. Very neutral, but frisky. There's one on ebay a bit over your budget, though. You could always make an offer.

https://www.ebay.com/itm/Black-Primare-I32-Integrated-Amp-Excellent-Condition-2500-RETAIL-USA-SELLER...

I've had some amps from Arcam, Naim, Creek, Rega before.
You know the usual suspects, the British stalwarts.

Now, the Anthem 225 sits in my main rig.
That should speak volumes on its capabilities. :)