My first recommendation would be the Audio Research I-50. Extremely musical and detailed. Gets everything right. Especially engaging. It is right on your budget. The really good thing about this is that it is competitive in the $10K integrated market.
If you can get into the $10K market via buying used, I highly recommend it. The $10K market offers strikingly better performance than the $5K market. I highly recommend listening to Pass. They have the fast response of solid state, with great rhythm and pace (although definitely not at the level of the Audio Research) making them really impressive.
There is also Luxman… I will admit to having high hopes for them and have been very disappointed at the trebly detail and poor midrange… the sound has not moved me at all. I have listened to their top level amp and a couple of integrated including the $10K integrated… to me, very disappointing. But that is me. I have been at this for fifty years and superficial flash will not get me.