Would a professional ASIC-machine holder eventually become a bank-like character?
Even if mining becomes more centralized, it would seem unlikely that miners would turn into banks: Their only advantage at shaping payments is that they may delay transactions by not including them in the block, so there doesn't seem a direct incentive to progress to banking.

On the other hand, payment processors and bank-like entities would have an incentive to foster mining efforts, as their business concept directly profits from a secure network.

So, banks should be interesting in paying for mining, but not vice versa.