I still think Sakurai views Smash as largely a Nintendo crossover, with much of the "Gaming's Greatest Crossover" rhetoric more being marketing language. However, the idea of a Nintendo crossover in Smash has expanded from "characters owned by Nintendo" to "characters associated with Nintendo".
This can mostly been seen in the third party choices that get into Smash. A general gaming crossover wouldn't go out of its way to get Bayonetta or Banjo, and it wouldn't lack characters from games like Doom or Grand Theft Auto. Likewise, the inclusion of characters like Sonic, Mega Man, Simon Belmont, and Hero instead enhances the Nintendo element of the crossover rather than detracting from it.
There are obviously a few exceptions to this. Snake, Cloud, and Joker seem to be the most obvious examples of this, but all three still have some degree of Nintendo connection (Ghost Babel, Twin Snakes, the first six Final Fantasies being Nintendo exclusive, Persona Q2, etc.). Plus, Sakurai played around with the idea of a Final Fantasy character more associated with Nintendo before settling on Cloud, so it's clear there's some acknowledgement that these are more exceptions than a rule.