I was going to paint it flat black. Would it stick?

If done properly.
Properly would consist of:
1 removing everything from the frame
2 Stripping the paint thats on it
3 ( I would sand it with some 300+ grain paper )
4 Priming It
5 Painting it ( I'd use something high heat, seeing as the frame is attached to the engine )

It would probably be easier to just take your frame to a local custom shop and have them powdercoat it   The one around here does frames for $100 if its already stripped

It would stick if you removed any trace of dirt,corrosion,grease,oil etc

Essentially the best way to do that is isolate the frame and strip it though

There's no reason to use high heat, the only place where there MAY be a heat issues is near the header pipes, but I highly doubt it'll be enough to wreck the paint...just get a good quality paint. You are more likely to have rock chips than heat issues. Try finding a wheel paint that claims to be chip resistant. I saw a few at Advanced Auto last time I was there. Prep work is everything...

I've never seen anything painted flat black that looked good, just so you know.  Try a semi-flat.  I'm particularly fond of the charcoal frames myself.

plus 1 million on the charcoal
see Dguyvers or my post....both have pics of fresh painted frames