To elaborate on my comment,
I think that any type of ‘reward’ will have a monetary value, whether that be indirect or not. Proposing to reward miners with something like ‘faster transaction confirmation’ (the example in your comment) would only add friction to their ability to reap financial reward. A savvy miner could expand their mining infrastructure, to gain a ton of quick confirmation credits. With that done, they could then sell faster confirmations to other users, and presumably profit in doing so.
I think that adding friction in this way is bad, it introduces external factors (the ability of a miner to network and sell their fast confirmation) that will affect the profitability of individual miners differently. Having an explicit monetary reward (ie the bitcoin block reward) keeps friction low: all miners that find a block will are equally capable of claiming its reward.
As I wrote above, If the reward is something to be sought after, then it will have value, even if indirectly.
There is no known way to create a secure cryptocurrency network without cost. POW requires energy expenditure, POS requires users to lock up their funds (opportunity costs), etc. Relying on users to pay these costs without reward makes for broken game theory, and a broken/insecure network.
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