That's quite annoying.
I've got the slimy stuff in my rear tyre, but i did find a very thin nail in it so it was the right thing to use in my opinion as the tyre's about half way through its life, and it will protect it now until i get my next one. It'll most certainly do the job for how i use the bike. But you're right in saying that although it remains runny it will only touch the main tread area of the tyre, or however much you lean the bike, but it won't really go anywhere near the rims.
I would be inclined to take the wheel to a reputable place, explain the issue with someone experienced in handling bike wheels and get them to take the tyre off, clean up the rim of any dirt or rust, then reseal the rim on refitting with whatever they use. Try not to let them give your wheel to a trainee though, been there, done that, and they generally mess your wheel rim up.
Probably cost you about £20-£25 at a guess if you take the wheel in off the bike.