If you were to ask me, I'm pretty much neutral regarding his chances. N. Sane Trilogy is a hot topic and the talk of town. Most of his games have been really, really associated with the PlayStation. With Ultimate becoming a gathering for major game mascots, I would say he is respectable. Sure people said that prior to NST he wasn't much of a talk of town type of thing, but hey, when has lack of games stopped BK?
In terms of this rivalry you speak of; yes. I kind of believe that this rivalry is a relevant matter of choice especially in the late 90s. You may say that people may only remember the rivalry between Mario and Sonic, but this was when Sonic was at the realest peak of his sensation. Crash simply stepped in while Sega was down for a while, and thus you had it. Hell, my parents were also in the moment at the time. We never bought a Dreamcast or Saturn, but we had a PS1 and an N64. In fact, when I browsed my parents' old collection of video games, most of them were either Mario or Crash, and this was the late 90s, so that says something.
To this day, my dad mostly mentions Crash when we're talking about cartoony non-Nintendo games, and he rarely brings up Sonic. Mom often questions me about his franchise about a couple times or two, and recently, my cousin (the same one that bought me Fire Emblem Three Houses for my birthday AND was the one whom I borrowed the Xenoblade 2 copy I played for a couple of months before returning it) bought Team Racing. Really says something if you asked me. Adding to this, me and my dad used to have Crash of the Titans and Mind Over Mutant on PSP, but mostly dad was the one playing while I was busy playing Tekken 5 DR or Tekken 6. Hell, me and my other cousin spent more time playing Mind Over Mutant when we weren't playing Tekken Tag 1 (largely because I just destroyed him with Double Ogres).
Point is, Crash is a character you really need to respect when it comes to console rivalries and Smash in general. It's something people who have been playing video games their entire life. Crash is very iconic, and even though I don't really support him these days, he's someone you need to respect when having him in Smash. Same with Sonic, Pac-Man, BK, name it. I'm not saying that he's this flawless figure that doesn't have any criticism in Smash, hell everyone was criticized at some point. But at the very least, it's something everyone should respect.