The Bitcoin system rewards miners for their contributions both now and in the future.
You see this all wrong. Satoshi needed to invent a system that was going to establish an immutable trustless permissionless decentralized consensus about transactions. It was not a matter of "rewarding people justly" or something of the kind, it was meant to make a system that solves a conceptual problem: making all participants AGREE upon:
1) the "just" creation of tokens
2) the "correct" transaction history to know who has the right to spend at any given moment.
Although this problem is trivially solved in any centralized, permissioned system (every bank does so), it is actually essentially unsolvable rigorously with the added desires of Satoshi:
1) immutable means that nor the rules of the system, nor the transaction and coin creation history, once part of the consensus, will ever change. (about the rules, he wasn't aware that the same mechanism that puts the transaction history immutability in place, also puts the rule immutability in place, because without the latter, the former doesn't make sense). So there must be a way that it is IMPOSSIBLE for the consensus to change its mind after having agreed upon something.
2) trustless means that one shouldn't assume about any entity in the system to want to be honest. In principle, all entities may try to be dishonest and screw the system.
3) permissionless means that just any entity can always decide to join and take up any role in the system
4) decentralized means that no form of hierarchy, leadership or whatever hierarchical decision taking mechanism should be in place.
5) consensus means that de facto, there is agreement on transaction history and rules. That doesn't mean that everyone is happy with that, but that such an agreement emerges from the system.
The last point is important, because it means that even "voting systems that give leadership to some entities" are excluded. This is why DASH, for instance, is not a decentralized system, because there's a voting mechanism and a leadership system to which voters can delegate their power.
From the moment that there is absence of decentralization, the consensus is simple: hierarchy decides what is consensus. Hierarchy decides upon what are the rules, how they should change and what is the correct transaction history.
Decentralization is needed in order to obtain permissionlessness, and immutability. Indeed, from the moment that a system can decide, it can decide to exclude members, not to include newcomers, and to change history and rules. So from the moment you have a centralized system, it is not permissionless and immutable any more. This is in fact the ONLY reason to require decentralization: to avoid losing permissionlessness and immutability.
With permissionlessness comes trustlessness. Indeed, if just any entity can join, then of course, there's no guarantee that they will be honest. Especially if it concerns transmission of value.
So, essentially, if you want a permissionless and immutable system, you're bound to include decentralized and trustless.
Well, there's essentially no strict way to make such a system. That is the Byzantine General's problem. What Satoshi wants to achieve, is impossible, strictly speaking. So something has to give in.
Satoshi's idea was that lying should cost you money, and that helping to come to honest consensus should reward you with money. However, it is unavoidable that you will make mistakes and that what you honestly thought was going to be the consensus, wasn't, so you lose money. This is why you should be rewarded with obtaining money.
The reward was also useful because Satoshi needed to print money, and printing money is usually frowned upon. So the idea was that money is printed to reward those that help to come to consensus.
Satoshi also wanted, in the frame of permissionlessness, that anyone would be able to join, and to help coming to consensus. So whether you already had money or not in the system, shouldn't stop you from joining. On the contrary.
However, you already see a problem. If you just distribute rewards to those joining and coming to consensus, what will stop their number ? As entities are anonymous, a single person can pretend to be a million entities, all coming to consensus. He would always get the rewards. So there needs to be a FAIR COMPETITION to come to the reward, one that costs the person value: proof of work. The idea is that you need to deliver proof of work to help come to the consensus, and get the reward.
That is how proof of work is used to come to consensus: to try to avoid Sybil attacks in competition for the reward, and hence also to make a cost for those making the wrong consensus decision. And this comes down to the consensus being ultimately determined by the largest proof of work.
The error being that proof of work being open to economies of scale, this brings in another form of centralization.
In fact, from the moment that consensus is rewarded, centralisation is unavoidable.
