It seems to me that my computer and its immediate vicinity get dusty far more quickly and accumulate a far greater volume of dust than anything else in my house. Am I imagining this, or do computers somehow attract dust more than other things, and if so, how?
The cooling fans actively draw in air(and dust) from the whole room. You might have airflow of 30-100cfm or more.
In general, because they have fans going all the time, or nearly all the time, that suck in air which includes dust. Dust particles are often electrically charged as well, which is why the dust will tend to stick rather than blow back out.
Has there been developed a dust-mitigation system (I mean an ongoing one, not just opening up the case every so often and blasting it all out with compressed air)?
Some cases have filters but I imagine there are still gaps that would let in dust.
You could seal the room airtight with a HEPA system for ventilation.
Might be a bit extreme but it would probably gain you much geek cred.
Not only that, but you have to remove the filter every month or so, wash it and dry it before putting it back in, or else it will completely clog up and either choke the case or let big clumps of dust in anyway. More hassle than blasting compressed air, and you have to do it more often ? Yay, progress ! 
I’ve built a PC that does not get dusty. I’ve had it for 6 years (internal HW has been upgraded several times, but the case & fans remain the same). There are two things to keep in mind to do this:
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Intake fans must have filters. I have a case by Lian-Li that has a washable intake filter. I don’t have to wash it often at all, but I have hardwood floors which helps. I wash it about once every 6 months.
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Total air volume being pushed into the case has to exceed total air volume sucked out. Standard cases have the opposite: they only have an exit fan inside the power supply so the case is under a slight vacuum. With this setup it doesn’t matter if you have filters, air will enter the case in all manner of places and suck dust in. On my setup I have 2x 80mm intake fans (filtered) and only 1 exit fan (power supply) so my case is under a slight positive pressure.
Man, you missed such an opportunity with this post.
“I’ve built a truly marvelous PC which does not get dusty, but there is not enough room to describe it in this text box.”
Not only does it not get dusty, it also runs Linux! <cue angels singing>