Off hand, I can't think of any FMJ .22 LR made today. I think you mean copper washed.
As for being grubby, yes, rimfire's get dirty. As for keeping them cleaner, I wipe them dry and then spray the mechanisms down with Rem-DryLube before I head out. It'll prevent a lot of the build up that the wet lubes cause. I give things a quick wipe down with a soft rag, a pull or two with a bore snake, a respray and then off to the races again every 150 +/- rounds or so. Before I put them back into the cases, I'll wipe them down again and another snaking but no spray as I'll clean them up when I get home.
Because I do clean and oil mine after use, I'll send a mag or a couple cylinders worth (usually cheap thunderbolts) into the berm at some object to settle things in (fouler's). I find it makes a difference, some don't.
One other thing I do with .22's is I'll send them thru my tumbler with corn cob for 45 minutes or so. It takes the waxy coating off. The coating is a sort of lube. After they tumbled, I throw them in a towel and roll them around to remove any dust then spray them down with the dry lube to replace the waxy crap. It seems a bit overboard but I can do a brick at a time. As I take .22's with me at every session, I keep a brick in each hard case (about 10) and a couple more prepped bricks on my going to the range shelf. I just replace what I use along with targets and such when I get home. I do the same to my .380's, makes everything super slick. Nope to the question does it cause more FTFs.
I'll have the tumbler out anyway as I first disassemble the guns and start soaking them, while that's going on I'll pop primers on the centerfires and send them into it with some walnut, start my cleaning, switch over to corncob then finish cleaning and putting things up.