Basically private key is use to store public key data into the wallet Right .
The private key is used to derive the public key, to sign transactions/messages and prove the ownership of the corresponding public key.
There is no other existence of private key while transaction. Miners validate the public key and then they will add new block to the ledger.
The private key itself is not 'stored inside' the transaction. But it is used to create a valid signature which basically says: 'I want to spend the UTXO from this public key'.
A valid signature can only be created with the private key corresponding to this public key. This signature can then be verified with the public key.
Miners do validate the transaction before including it into a block. After a block containing this transaction has been added to the blockchain,
every full node does validate the whole block for itself. This prevents miner from adding transactions with an invalid signature into a block.