I used to use the mesh strainer on the funnel, and, like it does for you, it would get clogged constantly. I eventually got fed up with it and just stopped bothering as I never noticed a difference in taste between those I strained and those I didn't.
I just use my hop bag and the rest just flows out of the kettle and in the fermenter. I only use the hop bag because the hop sludge seems to clump a lot more and clog up airlocks on active fermentations. Keep in mind that I don't use an ordinary hop bag, though. The 5 gallon paint strainer bags are much larger than your typical hop bag and allow the hops to move a lot more freely in the boiling wort. There's a bunch of threads on using them for this purpose.
I think whole hops are much more likely to clog your valve than pellet hops and/or break materials.