Why bother checking both public key and signature in P2PKH transactions?
1) The addresses would obsolete after one use.

2) If someone does a Sybil attack, s/he can prevent the transactions from relaying (which may also happen now), but also s/he can make a fake transaction from that address and relay it, as the attacker knows what the preimage is.

3) Even without a Sybil attack, someone could take advantage from eventual orphaned blocks and grab the money from all transactions in them, by making a fake transaction for each one.