Sub 1k integrated?

Details are everything. College, dorm, forget tubes. Unless you will be matriculating at Hillsdale, then I would say have your roommate build you a 300B. Typical liberal arts college, beer, there goes the amp. Or beer, there goes the speakers. If not directly beer then indirectly via beer induced volume cranking. 

Sorry, but it helps to be realistic. No amount of SQ matters once the voice coils start scraping and rattling. 

Klipsch Heresy's sound good, play loud, and fit most dorm rooms. Then if you can swing it, good SS integrated. That SA3 you mentioned will have enough power to drive the Heresy's nice and loud, but not enough to fry them when your roommate cranks em to 11, and will sound quite good. There's a lot of em out there, you can probably find some old and beat up for cheap. Might as well. If they aren't beat up now they will be anyway by graduation!

What a terrific picture you have painted, makes me want to go back to a dorm, ha ha!

But sorry, I was trying to be creative with the language, and being a scientist likely fell flat. To clarify, I am solidly middle-aged with three kids, hopefully who will all go to college at some point, so we are preparing for that financially. Hence my "good enough" and budget conscious approach to the hobby. I have been in love with stereo since my first mid-’80s Technics, and now with a bit of disposable income I am feeling like climbing at least one rung up the ladder. Still have the Technics which did witness the beer days of undergrad though, and it is faithfully serving as the garage rig with my old (vintage?) Bose 301 *edit* Series II’s.

And yes, Klipsch heresy IVs are on the long term aspirational radar.

Oh. Well in that case I do hope its Hillsdale.

Also good to know your plans. I tend to think long term when it comes to my system. This was 2004 https://www.theanalogdept.com/c_miller.htm  and this is 2020 https://systems.audiogon.com/systems/8367 A lot of the same stuff even over 15 years. A lot of new too. Pictures aren't sound but be assured huge massive improvement in sound. Slow and steady. Smart well thought out choices that last a long time beats frequent flipping. 

Anything you get will be head and shoulders better than your Yamaha AVR. The question is more are you looking for an anchor piece like you said or was that more colorful language? It really helps to know. 

Because a used amp like my Melody for example would be in your budget. And don't worry about tubes, with adult use tubes last many years. A typical amp like mine uses KT88, a set of which is under $200, and lasts a few years. Most guys change tubes because they want to not because they have to. The sound quality you get from tubes is unmatched with SS.

Any used tube integrated in your price range you will not believe how much better it will be than what you have now. Any. But if you really want an anchor there are some truly impressive integrateds in the $3-5k range that could be considered lifetime components. Or if not lifetime then long time like my speakers and turntable, or the phono stage in 2004, all in the 15-20 year range and all of which were at one time "anchor" type components for me. 

Another thing about long term, it helps to think of the whole system. If you're using hook up wire, patch cords, and freebie rubber power cords those are low hanging fruit ripe for upgrade. Something like a Creek that leaves room for better speaker cables might make more sense.  

What wire are you running now?

Running blue jeans now for speakers and interconnects

Okay well then you are on the right track. Except I would be looking at only pure integrated amps- no receiver, no DAC. Especially no DAC. The life span of anything digital is measured in dog years, one year in analog/tube years is five years digital, buying an amp with a DAC is like instant obsolescence. The idea of buying something good to build a system around is a good one, but for that to be true it has to be true long term, something digital simply never has been and never will be.