One point is that there is no reason for the entire secret to be one element of the underlying field, it can very well be a n-tuple of elements of a smaller fie

One point is that there is no reason for the entire secret to be one element of the underlying field, it can very well be a n-tuple of elements of a smaller field, with GF(2^8) being the somewhat obvious choice if you do not expect ridiculous numbers of shares, no need to deal with bignum math.