Best Integrated that you've heard under $5k

I have not heard them all. I'll stick with my old tried and true YBA Integre DT. 

Haven’t heard many but of those I’ve done serious seat-time with, these are the ones I believe are worth every cent!

McIntosh MA5200
Musical Fidelity M6si
Simaudio Moon NEO 340 IX
Luxman (can't remember model number but competitor to the MAC MA5200, at the time, around 4 or 5 years ago  and might have been a bit more than 5 grand. In A/B between 2 and 3 hours with Focal Aria 936 + Rega P3 with Ortofon Blue + Rega Apollo CD, the MAC & this Luxman were virtually indistinguishable from each other.)

Nice thread, lots of good suggestions here. For a super integrated the Devialet Expert Pro 140 (or 220 used) is amazing. Network streamer, DAC, phono stage, pre and power amp, will drive darn near anything. Roon ready. Amazing pristine SQ. I replaced a stack of separates, ARC pre, Hugo 2 DAC, Quicksilver monos, Pass amp, thousands in cables, blah blah blah, with one sleek device and couldn’t be happier.
Check out the S’phile and Tone reviews. 

I really love my McIntosh MA252. I got it for $3500 or so if I recall. Sounds way better than my old Hegel.

sorry, I didn't read the whole thing but your question is too vague.

Balanced or SE or does it matter?

and the most important... how much power do you need?

I use an Antelope Zodiac DAC that has analog inputs and the headphone output has plenty of power to drive my Avantgarde horns. It is awesome but for most people it is a non starter with the low power output, but goes straight to my initial point, your ask is way to vague to get any meaningful answers.