What if we were substitute the ANDs with XORs in some homomorphic encryption scheme like BGV(https://eprint.iacr.org/2011/277.pdf), LTV ( https://eprint.iacr.org/2013/094.pdf) ? It may be more efficient ? For example if we can substitute an AND with 6 XORs, we would achieve a better efficiency for some choice of the parameters.
As poncho comments, a homomorphic encryption scheme that only supports the XOR operation would not be fully homomorphic, i.e., it would not allow evaluation of all functions. However, you can try to optimize the implementation of a function in order to minimize the number of AND's at the cost of extra XOR's. This is not only relevant for FHE but also in secure multi-party computation (MPC), where the XOR operation can also usually be computed much faster than AND.
Optimizing the circuit of a function to include less AND's, and possibly more XOR's, is a well known technique to make FHE and MPC applications run faster. In fact people have made special compilers to automate this process. An example of such a compiler in the MPC space can be found in this paper (although by now there are probably better ones around).