WOT KOD Spoilers, so if you haven't read it and don't want it ruined, skip this one.
Rand gets his hand burnt off. He has no emotional reaction whatsoever. We also see that the Asha'man are all trained to not let their emotions affect them at any time. These men are basically required to be heartless killing machines. And soothing them wouldn't work. Based on rand, though there are likely other methods, men have virtually no emotion while channeling. When they do feel emotion, they promptly rid themselves of it via the flame and the void. Women need to ignore their emotions to channel, not get rid of it. So if the channeler is competent at ignoring emotions, the rioting wouldn't really work. The mistborn would have more of a chance against an Aes Sedai than an Asha'man, but honestly I see them charging and getting ripped to shreds.
The other thing we have not specified, in said fight, do the Mistborn/Channelers understand opposing magic systems? That would make a big difference in how well they fight. For instance, if a mistborn has no idea how channeling works, the whole electrum to see where your head gets chopped off thing doesn't work. If the channeler doesn't know how allomancy works, they are likely dead from metal on their bodies in seconds. It makes a difference. If they both know, however, they can prevent attacks much better.