The issue is then boolean functions do not form a polynomial ring with this "sum", because unlike XOR, OR is not a group operation - it doesn't have a well defined inverse.
This is an important distinction between whether we are preserving structures on rings or on a group. An example would be mapping positive to negative numbers which is a group level transformation but not structure preserving for order while still being invertible
Perhaps you meant to say that AND and XOR are products and sums.