You have gotten some great advice so far and your thinking of the two only ways to get rid of the flying rats; building a Fort Knox tight coop and run, and removing their access to the feed.
Of the two, a treadle feeder is the cheapest route and yes, they start at $85.00 plus shipping for a good American made one and can run twice that but those are the ones you have been warned about; the guillotine style feeders. That said, even a really poor treadle feeder like the Rent a coop or Grandpa feeder can deter sparrows. The problem is their cost, twice what you need to spend, the fact that they are made in China, and the three week training regimen that requires keeping the feeder blocked open which teaches the sparrows where the feed is, how to swarm the big treadle plate, and those style of treadle feeders do not stop pigeons or rats or mice in many cases if you read their negative reviews on Amazon.
Costly, yes, but if you keep it up out of the poop one will last for a decade or more and if you are feeding sparrows you are already paying for the feeder without having the benefits of having one. Not only is that sparrow poop awful, it is packed with disease plus the mites, lice, and lord knows what else a sparrow can bring into a flock of chickens.
BTW, price isn't always an indication of quality. You have Chinese made feeders like the Grandpa and Rent a Coop guillotine style feeders that are twice the cost of an American made feeder. And there are other Chinese knock offs of the Grandpa feeder that aren't any better or worse for half the price but they incorporate the old design that is unsafe due to the overhead lid, has the three week long training period with an open feeder, and all the mechanical design flaws, pinch points, and the lack of a soft close version if that matters. Plus those feeders are notorious for not having spare parts available and treadle feeders do have wear items that will need replaced.
So do your research on the treadle feeders. Avoid the websites that link back to Amazon or a manufactures website because those are commission sales articles, not independent reviews. They are supposed to say when they are getting a commission but most do not. Read the reviews and the instructions before you buy and pay close attention to the negative reviews.