Look with a magnifier and very bright light. That screen *is* very fine, and is probably causing the bad needle settings, and is effectively stopping fuel flow to the main metering needle adjacent to it. If you have acetone, try a cotton swab... repeated applications.... whatever is in there probably was stopped by the screen (good thing) and now needs to be removed (not so good thing).

If you absolutely cannot remove the crud, go to your local saw shop, get a carb rebuild kit of nearly any flavor. It will have the screen in it. Remove the old with a very sharp, pointed scribe being VERY careful to not push further down, and not damage any surfaces. Press in the new screen with a short piece of 3/16" wood dowel, clean, you're probably good to go.

If you do all that, and it still will not needle properly, new carb time. Trust me on that.

You are using at least "dual-filtering" (gas jug) and tank klunk, right?