No, that's a false dichotomy. And also the "lucky" is relative to the whole scheme. The value to the miner of finding a hash to make another block has gone down and will continue to go down, that's partly why transaction fees are so high.
In any case, that's like saying someone is "lucky" to win a video game in Dave and Buster's. That's not the only way to incentivize validators to timestamp transactions. All you really need is a consensus protocol.
Ripple for example has a consensus protocol that can be run by an entire LOCAL community and can fund itself and the resources it uses. Without requiring a global blockchain. And Bitcoin validation is effectively centralized in the hands of a few miners.
The PoW isn't meant to elect the next miner. It's meant to ensure the integrity of the chain and maintain a lead in the work cost for the honest chain.
Transactions are already signed and can't be forged. The only reason for the chain to exist is so that everyone knows about every transaction, to eliminate the double spend problem.
To timestamp transactions in a distributed way.
Transactions can be signed, proving authorship.
At signing time, you can prove the transaction happened AFTER something else.
The only thing missing is PROVING THE TRANSACTION HAPPENED BEFORE SOMETHING ELSE.
For that, you need an incentive structure to keep each transaction be accepted by someone, somewhere, in a growing merkle tree.
That's the blockchain.
However we don't need proof of work to elect the next miner for every block. It leads to an incredibly wasteful arms race.
In fact we don't even need every transaction to be verified by a miner. Only the merkle tree that happened before the block signing time.