Perhaps this is why Saki is gone. There wasn't really a reason to remove her since she was in the last game, but if some or all of them needed a bit of an update so that they don't fall off the new stages too easily, and she wasn't gaining any traction, then they could have just decided to drop it. Probably the only explanation that isn't arbitrary (though to be fair, it could actually just be that they decided not to).
As for the other ones that have been in the previous two games, but not Ultimate, I'd imagine Lakitu and Spinies, Infantry and Tanks, and Kat & Anna were removed due to the new Assist Trophy mechanics. You'd have to do more spritework for the the former two, and the ability to hit them would make them kinda worthless anyway (don't need 3 Starfys). Kat & Anna would be pretty ridiculous if you tried to hit them since they're so fast, and they would probably instantly die if a Smart Bomb went off or something since they span the screen.
As for Tingle...I dunno. Maybe it's so that he's not confused with the stage hazard that has at least the balloon part?
EDIT: To be clear, I know that Assist Trophies turn off by stage, but having the character be inactive on Great Bay, but also do something similar as a stage hazard could be a concern, though I don't really know by how much.