What is the reason to pay miners "monetary rewards" for mining process? Is it correct to say that:

In Bitcoin, it is crucial to incentivize miners by monetary reward, because mining process has significant monetary cost (energy/electricity cost). ? otherwise, mining process is not affordable for miners ? is this sentence correct? 

I mean if we assume that mining process to solve PoW was free (consider accessing to free electricity was possible for everybody in earth!), then could we have fee-free transactions in Bitcoin? and incentivize miners by a non-monetary rewards? 

In other words, the reason of using monetary rewards is that consuming electricity has monetary cost for miners and so this monetary reward is compensation of miners electricity cost ?

Please note that my purpose to ask this question is not the reason of PoW as consensus mechanism, but also I mean the reason of using monetary rewards not reward. We can have other types of rewards, not necessarily monetary reward. But apparently in Bitcoin because of nature of PoW that has monetary cost for miners, we may have to use monetary rewards for miners.The matter of my question is not reward itself, but also the type of reward Bitcoin uses that is monetary reward.

I'd like also to mention following notes:

(1) non-monetary reward does not work for PoW, since mining process has monetary cost for miners and so non-monetary is not affordable for miners.

(2) the purpose of non-monetary reward is bringing fee-free transactions, since resource of monetary rewards is transactions fee. 

(3) we can make non-monetary reward non for selling such that a point for faster confirmation only belongs to a specific public key exclusively and it is not Non-transferable to other public key.
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.