What's the best integrated amp in the $1500-$2000 range?

Depending on your speakers and since you are willing consider a pre-owned amp, may I suggest an YBA Integre Alpha? 

I had nearly exact question several months ago: https://forum.audiogon.com/discussions/seriously-considering-tube-preamp-opinions?page=2&sort_order=desc

Perhaps some answers in that thread will help. I was very tempted by the Vincent 237mkII but no phono, which you said you wanted. Many, many people also suggested Parasound.

The Rogue is a great call, I suspect. For the money, and depending on power needs, the QS Integrated is an unbelievable value -- it's $2k, new, so perhaps a non-starter, but if you like vinyl and would go tube... http://quicksilveraudio.com/products/integrated-amplifier/ I heard this integrated drive some Dynaudio Evoke 10 speakers (84 db) and it did very well -- lovely, organic, dimensioned soundstage and tonal balance.

I’ve heard nothing but good things about the Rogue line but have no personal experience with them.  I got a PrimaLuna Prologue Classic last year and have been deliriously happy with it.  It cost me just a tad under the top end of your budget....one needs to bring a phono preamp (Actually, a built-in was a special order option which had pretty good user ratings.  This unit has been replaced by the “Evo” line so not sure as to availability as new.). As has been stated, a lot depends on the speakers you have.  I’m running JBL L65s, from the mid-late 1970s and they’re in the 89-91 dB sensitivity range....

If you want something with a phono stage built in, then you should be able to find a well cared for Peachtree Nova 300 in your price range.  It also has a very nice DAC.

You can look for used WLM Minueta push-pull.

or Luxman SQ-N100 with MM phono stage and remote control, there is even a headphone amp. This one is probably best buy if you like tubes. 

Owned both, very nice for the money.