Some coatings, especially elastomerics, are designed in part to seal leaks. I'm talking pinholes, nothing major. But yes such coatings do serve this purpose.
As for the original question, if you want elastomeric on it, don't coat it with asphalt now. Asphalt is one of the hardest products to coat over on a roof. You can do it. Acrylic DTMs work and Elastomerics work better than anything else over it. It's still an iffy proposition though. Any asphalt coating you put on needs to age a fair while to weather the solvents out of it before recoating with an acrylic product. No paint company is going to stand behind their product if you use it this way.
I'd wait until spring to tackle it. Once the weather is good, scrape down and if possible power wash as much of the old coating off as you can get (since it's already disappearing.) If you have any major rust problems, I'd hit it with a coat of Ospho or similar product. This is an acid wash that will chemically neutralize the rust. Then I'd prime it with a red oxide DTM primer to avoid future rust issues. Patch any spots you suspect are leaking or you expect to leak with a good quality elastomeric patching compound and then put a couple coats of a good elastomeric roof coating on there. That's the full scale treatment I'd recommend. It may seem excessive, but if you do it right to begin with you won't have to touch that roof again for a very long time.
For this type of stuff we usually use product from Richards Paint company. www.richardspaint.com
They have some very good elastomerics for less than many of the other big brands.
290 Roof Shield Thermal Elastomeric Ceramic Roof Coating
P830 Seal-Flex Smooth Brush Grade Elastomeric Patch