While you would only need to edit the save to add cloud to the list of people allowed to be placed in the PHS to enable the player to have a party without cloud. there is a small problem with the way this works in stock FF7. the game only allows the player controllable model to be cloud, tifa or cid. this applys to both the field and world maps areas. the games does not crash just if you don't have cloud, tifa or cid in your party you see no model.
This is partially correct. I tested this a long time ago (check my post history or something if anyone is interested) and in scenes where a character has a path mapped, the last party member swapped into the party will be the controllable one. For example, in Cosmo Canyon you're able to control any character in the first screen of the town, but if it's someone like Yuffie they'll start off at the candle. When you try to go into another screen they'll be invisible.
Occasionally you can control the character even when they're invisible, and occasionally when they're visible they won't be able to trigger the transition between screens (meaning you can enter a town and become completely stuck. This happens with Aerith in Kalm.) It's a pretty messy area to be honest, it seems to work differently in almost every place.
Also note that having Cid or Tifa in the party doesn't mean the game will default to them being displayed on the screen (except on the world map, where even if none of the 3 leaders are present Cloud will appear). The exact mechanics aren't really clear. Battles work flawlessly without Cloud but this kind of thing will need some serious field editing to work. It'd be an excellent mod but a lot of tedium would be involved.
In short, don't untick Cloud's "unswappable" box in Black Chocobo unless you've backed up your save and can be bothered with experimentation.