Anyone critical of the current mismatch between speakers and amps forgets that standard audiophile advice is to max out on quality of speakers first and then work on what’s up the chain. OP did this and is asking the next step. Bravo!
I would suggest that the best advice would not include a focus on a specific model or brand yet. With a limited budget (even though $5k isn’t chicken feed) I suggest that used gear is the way to maximize the value you get, and the best way to try different combinations of gear is to buy used and listen to them for extended periods; keep what you like and resell/replace what you don’t. To do this, you need to establish a range of brands and models that you're interested in, and be flexible about what used stuff is available at good prices.
So that’s a perpetual journey or at least years. But it’s fun. Buy good quality gear from here, from good dealers, and even ebay and Craigslist (I’ve had plenty of success on both). A modest percentage of your budget (say, 10%-20%) will be in losses in buying and reselling gear, but that will easily be balanced by the tremendous value you get in buying used gear.
Look for brands like McIntosh (rarely a bargain though); Levinson, Krell, Classe, Pass, the others commonly mentioned here.
I’d focus on separates for now because ultimately that is likely to sound best, but also because you can test variables more easily with separates than an integrated. On cables I just would avoid the $5 kind for now. Until you settle on gear that you’re going to keep a long time, you might find more cable mismatches than matches.
With a limited budget, I’d stay with solid state. Tubes are great but they can be expensive and tube gear fussy. And if you are swapping through gear looking for the best combos for those fine speakers, you won’t want to have spent big$$ on tubes that you can’t use when you change gear.
Good luck! This is part of the fun of the hobby; don’t look at it like a leap and a stop...look at it as a series of fun Saturdays!
