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.
Proof of work means that the lucky random miner has invested energy on a previous valid block, and is lucky enough to find another valid block.
If you take that away, you could as well use a database and a trusted timestamp server.