Well in theory the base math is indeed the same; unfortunately though the "randomly chosen" part of shamir's secret sharing is fairly important to the security because information theoretic security of the scheme requires each fragment to be as large as the original secret by way of essentially including a desired count of random data blocks to the original before applying the reed-solomon-like erasure coding to it where now enough fragments to reconstruct the secret plus all random blocks have to be combined. Also the way of usage of the erasure code has to be selected to not be leaking information but that's more of an issue of not picking a bad way of how to implement the basic concept here. Basically just a case of "do follow the instructions to shamir's secret sharing, don't do something different just because it's a popular way of implementing reed-Solomon".

Yes, you can just GF(256), but if you're worried I'd also just use a prime field instead.