How is mining associated to blockchain transactions? I know how mining works and how transactions are added to the blockchain. What I don't understand is the association between the two...if any.

I've read a few times about how a cryptocurrency transaction is added to the blockchain through consensus. This often means PoW. Then there is mentioned that miners perform PoW and the transactions are cleared. How?

Miners mine for coins right? What does mining have to do with verifying unrelated transactions?

At least for bitcoin, is it even possible for miners to keep up with all of the transactions occurring?
How users / nodes see it:

When transactions are made, they stay in nodes' memory pools ("mempool") until a block that contains them, becomes mined and relayed. When this happens, that transactions become a permanent member of the blockchain (unless that block is orphaned).

How miners see it:

Miners' major incentives are block rewards. However, to maximize their profit, they can also include transactions in their block (According to BlockChair, 124k of 510k blocks include no transaction.) If they include transactions in their block, they'll be able to collect their transaction fees. That's why most of the blocks include transactions (and the miners who don't include transactions are mostly illogical).

tl;dr When miners include your transactions, your transactions become a permanent element of the blockchain. They do this to collect your transactions' fees, to maximize their profit.