I don't think a transaction would just become valid if a miner (who mined a block) signs it. Each participating node will independently verify the chain for sig

I don't think a transaction would just become valid if a miner (who mined a block) signs it. Each participating node will independently verify the chain for signatures of transactions before it accepts the latest block. So a miner who mines a block with invalid block will just waste his efforts on it. The chain length is used a proxy to find out a valid chain only when two competing chains have valid blocks throughout but have different lengths.