Best vintage amps (late 70s / early 80s) under $3k?

Not really a practical option as these are extremely rare, but if you happen to get lucky and find a SONY TAN 8550 V-FET amp, grab it as fast as you can! Good condition one will cost $1500-$2000, only challenge is getting replacement V-FET if you blow one. It's in a different league than most of the amps mentioned in this thread, and I've auditioned many of them back in the days.
If you are into tubes, somewhat less rare are the Quad 2 monoblocks. Make sure they have the original transformers, all the rest is easy to replace.

No mention yet of the Quad SS amps: the 303, the rare 50 monos and the 405. I have 'em all - and they have been restored/upgraded! Historically significant circuit designs from Peter Walker!

Here's a real sleeper: the Brown Electronic Labs BEL 1001. Designed and hand-built over the years by EE Richard Brown at his home in California. I have his very first one (serial no. 0001). A favorite of HP's at TAS around 1980 and considered reference quality! 

I can’t speak to everything, but. My Robertson 4010 has been upgraded, and it collects dust. I have a Stasis 2, I replaced EVERY resistor, every capacitor, the bridge rectifiers with hexfreds, and had Jon Soderberg install matched transistors in both channels, as well as PS caps. When done, he said it was the best sounding Stasis amp he had ever heard. I have forgotten all the names of all the amps which I have owned or listened to. Based upon heresay, the Levenson 23.5 is killer, Jon Soderberg also spoke of how some Levinson amp he worked on controls  speakers. From those I have or have heard, I prefer the PS Audio 200C or 200CX, IF you can find one that is working. They REALLY need to have the bridge rectifiers replaced with Hexfreds, and a recap would be a must. Their slew rate is something like 300 or 400, that’s speaker control! I would also have all the electrolytic caps bypassed with really good film and foil caps. I have yet to try them, but these are quite intriguing;
https://www.partsconnexion.com/mundorf-m-lytic-hc-mlhc-2-pole-electrolytic-capacitors.html
They practically have to give these away because they can and likely will catch fire if you plug or unplug the outputs, and possibly the inputs before you turn it off and give it a minute to discharge the caps. You ought to be able to buy and upgrade this for well under 1.5K, leaving you plenty of money to play with. Interestingly Stan Warren, the S in PS Audio, started Superphon about the same time as the 200C came out, and they are giving them away as well. You can do pretty much the same upgrades to it as the PS Audio amp and be well under 1.5K. These 2 amps are real sleepers. The problems that the reviwers had with the 200C are largely due to the diodes. These are giant killer amps. I am sure there are others, but these really impress me, and can drive my Thiel CS-7 speakers. If you go that way and need help, let me know!