Why will a miner verify the digital signature, instead of directly adding a transaction to the Merkle tree?
If a miner would do so, and the transaction they include is invalid, they would produce a block with invalid transactions in them. As all full nodes on the network (not just miners) verify all transactions in blocks, they would reject this block, and the miner would lose their reward.

By extension, other miners will avoid building on top of such a block, as they too do not want to lose their reward.