Brown algea will go away in a cycled tank, but not a betta bowl that gets 100% water changes.
I would just keep cleaning the bowl and get any kind of gravel/decorations out in the meantime. It'll stay attached to anything that doesn't get cleaned thoroughly, including some on the betta itself. Keeping the water changed and the tank wiped clean will evetually get rid of anything that may have been left on the betta, and then the cleaned decorations and new gravel (can't really clean gravel all that well) can be put back in.
I have seen ghost shrimp pick at this stuff before, but they really can't do much to it. As for leftover food, yeah, they work great and hardly make a dent in water quality. The only problem is that they're impossible to remove when doing water changes. I usually fish five or six out of the buckets when I gravel vac, and even then they can fling themselves over five feet when out of water. (The cats really like ghost shrimp as snacks, so I barely ever get there in time to save them.

)
Snails aren't a good choice as any type of apple snails needs 2.5 gallons,
at least, of their own. They are great pets, but they're not to be used as a clean-up crew. They are WAY too messy. You'd have to do water changes at least once a day to keep up with their waste production when they're fed correctly.