recommendations for amplifiers that don't emit tons of heat

tripg- you read into the analogy correctly. We have an A'gon comrade with Anthem I225 Integrated amp. It's rated at 225 watts into 8 ohm/310 into 4ohm. He's running Paradigm Studio 100 v3. The speakers do not flinch or suffer any driver compression. He works the amp hard and it does not get hot. And its not sluggish. These are in your pricepoint - new with warranty. Now, are their more transparent amps out their? Of course. 
https://www.anthemav.com/products-current/collection=performance/model=pva-2/page=overview

https://www.anthemav.com/products-current/type=amplifier/model=mca-225/page=overview

Ayre also makes some great 5 series amps. 150 watts into 8 ohms. They should run around $2250 - $2600 used. Very transparent. 

Regarding larger amplifiers my Son of Ampzilla mk2
Is very fast. The late great James Borngiano was a great designer.
This amp has a Huge 2,000va transformer for control and I believe 12 bipolor
Transistors per channel and over 100k in capacitance.  And grounding eliminates ground loops in this amp low level detail is very good also.
The matching preamp is very good also ,but I have a custom Lundahl transformer based 4P1L - DHT preamplifier Which being single ended really pulls the music out. TIP OF THE DAY - Buy Stabilant22  to put on all Audio connections just one mating side very thin coat,first clear everything with Isopropyl Alcohol 99%
If possible.lasts for years and by far the most technologically advanced on the market . If it's good enough for NASA,then I am good with it also. I use on tube pins and everything else give it a solid 30 minutes to dry, great stuff !!

If you want tubes, Primaluna amps running pretty cool.

Y'know, if you like the sound you are getting there is a much less expensive option, which is simply to have a vent installed to the outside that allows the heat to escape the room rather than warming it up. This can be done for a few hundred dollars at most (which if really sophisticated might include a wall switch with a fan to help move the air). I've seen this be very effective!

Most any Class D amp is relatively small, very power efficient and runs cool. Many good ones out there. I use a Bel Canto c5i in an enclosed audio cabinet and also have other BC 500w/ch power amps. Neither ever gets even slightly warm.