Sonic and Snake would've made it into Melee if there was enough time. 3rd party video game characters were already a fine idea after Smash 64. It just was impossible at the time. No floodgates were open to begin with.
Sheik was added before Zelda was to Melee overall. She didn't start "anything" and is an outlier(she also is a main character in that game, which is one of the biggest Zelda games in existence. It's still beholden to that. On the other hand, ALTTP, the other massive game, still can't even get its main villain a cameo in Smash. There's nothing to really put into this kind of logic here, as it means nothing). Dr. Mario was in mainly because he was a clone option. Spin-offs weren't given some special treatment there. Also, keep in mind that only Mario Kart even got something via its own music section(due to necessity) and its own stages. We don't generally get spin-off content for the most part, with very little when it comes to Spirits or other similar cases even then. Like, Tingle's specific game content is not that much. Hell, Sonic is one of the few third parties to get lots of spin-off content with Stickers, etc., but even then, those spin-offs are part of the normal game continuity, something that doesn't work with other franchises. Zelda's spin-offs are at best their own thing, with at most Link's Crossbow Training maybe having no issues with potential canonicity(since it doesn't contradict or do anything odd. It's just a silly game at best).
If we got another one-off, it'd be because they're part of the main trio. Which is the only reason why Sheik made sense. She is freaking Zelda. Also, keep in mind, we were likewise intended to get Toon Zelda before, but also a Smash OC in Toon Sheik(and even if that became Tetra later on, her appearances were a one and done game too). Clearly being one-off itself wasn't that important. It's just they weren't the main Triforce trio, which has proven to be a lot more important than anything else by now when it comes to being playable. Even in cases where the Triforce doesn't exist within a game, it doesn't change that that version of Zelda, Link, Ganon, or Ganondorf are looked at.
Besides? Nothing has stopped Sakurai from adding a spin-off character in some form(again, more or less). He simply hasn't. There could be many reasons. Lack of importance, licensing being more difficult as an option, a focus on different contexts, etc.