shine allows for all that stuff, but so do a bunch of other moves the spacies have
they can jab to grab, nair to grab, drill to grab/utilt, probably bair to both up smash and grab, uair to more uairs, utilt to grab/upsmash/uair/nair/anything, etc.
realistically, i dont know if fox would actually suffer that much without shine. i know iori has won a tournament using fox to beat xif in the grand finals and didnt shine once, the videos exist too but they might be hard to find
falco's combos seem to rely on shine a lot more than fox really does, but he also could probably do a lot of dair utilts as well to continue stuff
space animals would just have to think and mix up their shield pressure a little more, alternating between spot dodges, aerials to up tilts or jabs, spaced aerials, empty jump grabs, etc.
fox might have trouble edgeguarding other space animals though, he wouldnt really have an answer to them coming from below the stage except dsmash which gets teched too often