I don't see how this makes any sense at all without making additional assumptions which would make Bitcoin pointless. The stochastic nature of mining means that there is always large uncertainty about block delivery times— the block gap is frequently anywhere in an hour window.  Paying non-trivial amounts for trivially copyable data to get it seconds faster when the common arrival uncertainty was already an hour does not generally make sense for most applications.

Keep in mind that a little payment isn't sufficient, mining needs to be expensive enough that an attacker can't do a short term attack— doubly so in the future because there will no subsidy motivation for honest behavior anymore, the motivation to behave honestly would be fees and preserving their value— Without the proof of work Bitcoin is just a particularly computationally and bandwidth expensive rumoring network totally vulnerable to network attackers.

You might be assuming that the miners would keep blocks secret from parties that don't pay them, but this can only be done if mining is centralized since otherwise a miner who promotes their block further faster gets built on and they become the valid block.  Moreover, block secrecy violates the less discussed second half of the Bitcoin security assumption:  Bitcoin assumes that honest parties control a majority of the mining applied computing power _and_ that information is easy to spread and hard to stifle.  The latter part is critical because you can only converge on the longest chain if you've actually heard of it. We don't talk about it much because it's not hard to meet today and you have bigger problems if you can't even get the blocks... but it really precludes any kind of paid access to blocks from being viable.

As far as the OP's issue— it would be pretty easy to add a trickle mode to the client where it only connects to the network periodically and only fetches blocks just to avoid failing behind. So you'd leave it running... but it would only connect to a few peers on the network ever half hour or so, pull new blocks and disconnect, then process them slowly and unintrusively. But I don't think there is enough demand for such a thing to bother coding and testing it.  Maybe that sort of thing could become the default behavior for minimized gui full nodes that aren't listening, but I dunno— differences in behavior create confusion.