I cut a hole into a small plastic tub and then buried it in the substrate so it looks natural, but I have to ask, how vital are Hides to Redfoots well-being. Are they needed or could I take it out without any ill effect to them?
They're needed. It helps the tort calm itself if it gets scared or worried about being in the open are being handeled by a creature at least 20x its size (it thinks it's going to get eaten...). Without hides your tort will be more stressed. With hides, he'll be more likely to come out of his own accord and be more friendly with you, since he knows that worst case scenario he's got a safe place he can retreat to.
Wild tortoises, even Red-foots, use hides as a secure place to sleep, digest, spend the uncomfortable parts of the day, etc. without having to worry about predators, bad weather, etc.
IDEAL hides, which are tough to do, would combine materials that the tortoises can snuggle into, and a certain snugness on top as well simulating burrows, etc.
I personally like offering several hides- warm, cool, humid, not so much. My current hides are slabs of natural or cork bark propped up a little and lightly stuffed with fibrous moss (that I really don't work to keep moist- instead, they are built on top of a layer of cypress mulch with heating cables in it, and I just pour water into the cypress. It creates lots of humidity that is trapped and concentrated in the hides.
Important, but not as important to other species like yellowfoots or hingebacks. RF's are normally tropical savanna tortoises, but they still need a hide or two in the enclosure. An easy way to make a hide is get a 15 gallon Rubbermaid, turn it upside down so the top is actually the bottom, then cut a hole as the entrance and fill the hide with sphagnum moss and leaf litter. You could also give them a sense of security by making half the enclosure nice and dark, and give them a substrate to burrow in.