I’d replace the speakers first rather than trying to put a bandaid on them with tubes, and you probably don’t wanna run your Matrix 3s with a tube amp as they generally require some power to get the best out of them. Those older metal tweets are likely the biggest source of the brightness you’re hearing so fix the source of the problem rather than jumping through expensive hoops to get them to sound better. Given what you’re looking for I think you should give the LSA-20 Signatures a good look — they are available here direct from the manufacturer for only $1699/pr. They’ll definitely add more bass to your system, and their silk dome tweets will very likely tame your brightness issue without sacrificing detail. Read the reviews.
After that I’d upgrade the preamp as it’s the heart of your system and greatly affects the sound of your overall system or, even likely better and more cost effective, get good integrated that’ll upgrade your amp at the same time. Then I’d do a good DAC, and then I’d wrestle with cables. That’s what I’d do FWIW.