I've found this characteristic to be HIGHLY dependent on the woods.
In particular, IME(!!), wenge/wenge necks seem to really accentuate string death, whereas something with a maple neck fares much much better. My current ash/walnut ash/maple MTD sounds wonderful for the entire time the strings have lost their new sheen but still have a decent roundwound ring to them. Further, it doesn't sound bad to me even if I were to let the strings get kinda' thuddy, though I rarely let them get that bad because dead notes drive me nuts (which is why I'm super particular about dead zones on basses I buy).
Strong IMHO on this, but the tulip/wenge 535 I had just sounded horrible for anything but slap once the strings got past a relatively new condition.
For me, SS strings (pretty much all I use on my 535's) have three distinct phases they go through. The brand new brilliance/sheen phase, and then a MUCH longer lived "alive" but much more tame on the high end phase, and then death.
One of the things I really like about SS's is, if you don't mind dropping a treble control some when they're new, you can get a TON of life out of them, and that much longer lived middle phase just sounds great to me... almost as much brilliance as a brand new set of nickels, and way way more string life than them. You'll never get SS's to feel as nicely slickery as nickels, but I'm used to that anyway. I just can't get a set of nickels to last for me at all.
A buddy of mine had an ash/maple maple/rosewood 535 that just NEVER seemed to sound "bad" no matter how dead the strings were. Same for an ash/maple maple/ebony 535 I used to have. My current ash/walnut ash/maple bass is almost that good, but not quite.