These are all great suggestions with your theories confirming ours - that it's likely the carb. We've never removed one before so it should be entertaining, but I'm taking to heart the fact that it doesn't seem to be a difficult job.
Would it be better to use the spray cleaner or should we let it soak in the other stuff?
You can try mixing about a 50% mix of gas & cleaner. Run it, let it set for a day or two, you might get lucky.
I have used a combination of spray cleaner , thin wire and compressed air. Never had to soak the whole carb but it might help if the first attempts don't work. I have used Berryman's B-12 fuel conditioner/ cleaner for many years. Other products may be as good. Be careful, Berrymans can remove paint.