Why bother checking both public key and signature in P2PKH transactions? In a P2PKH pubkey script, first, the public key provided by the intended spender is verified to be correct and then this key is used to verify the spender's signature.

I wonder why just verifying the public key is not enough. The pubkey script contains a cryptographic hash of it, so reversing should be impossible and anyone wanting to spend the UTXO associated with that script needs to have the correct public key. Doesn't that already authenticate the spender sufficiently? I understand that signing is necessary in P2PK transactions, but with P2PKH, hashing and signing seems redundant to me.

Isn't mere hashing of the redeem script also used for authentication in P2SH?
The pubkey script in P2SH just checks if the redeem script hashes to the correct value and then it already executes it. If hashing is taken to be secure in P2SH, why not also in P2PKH?
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.