But wasn't Naughty Dog owned by Sony when Melee came out? Correct me if I'm wrong.
At the time, While Crash Bandicoot was in a 3-Way partnership with Sony, Naughty Dog and Universal Interactive, At the end of the day the IP has always belong to Universal since Day 1.
Crash Team Racing (1999) was Naughty Dog's last game for the franchise, I assume their contract with Universal was up and then for whatever reason Sony and Universal got together with Eurocom/Cerny Games for Crash Bash (2000) the last Crash Bandicoot game for the original Playstation while Naughty Dog moved on to PS2 development.
Sony bought out Naughty Dog in 2001 and while Crash Bandicoot is heavily associated with Sony's Playstation brand, I assume Universal thought that a big brand and IP like Crash would sell like hotcakes going multiplatform and of course decided to go that direction, But the quality and the sales of the franchise also declined.
Banjo-Kazooie missed out on Melee and he was owned by Nintendo, But despite Rareware being under the Nintendo umbrella, Sakurai stated that he would have still been difficult because of the language barrier and Rare being a western studio in the UK, If someone like Banjo couldn't have made it in, I kind of doubt Crash would have had a shot either.
Sonic the Hedgehog was also Mario's main rival and had the advantage of being owned by Sega another Japanese company which would have made communication much easier, Grant it who knows what was Sega feelings and relationship with Nintendo back in 2001 seeing how they had to "Tap Out" of the console wars.
I imagine Crash would be in the same circumstance as Sonic, But much more unlikely at the time since just like Banjo-Kazooie he was a western character and despite going 3rd Party at the time, We don't know how closely Nintendo was willing to work with Universal considering how they were working very closely with the enemy that is Sony.
Luckily for us Crash is back and his chances are as good as any and better than ever to join Super Smash Bros Ultimate.