The problem would be still to maintain the UTXO set. Fortunately, by using a two-party dynamic authenticated data structure, it is possible to have only miners(and also powerful full nodes willing to hold full-state, like block explorers etc) to store the whole set, while others can check proofs of set transformations included into blocks, storing just 32 bytes. For details, see introduction to the paper https://eprint.iacr.org/2016/994 (and blockchain processing benchmark graph, 2nd half of pg. 14)