Yes, people are more likely to be successful in a suicide attempt if they own a firearm, of course, that is a given, the firearm makes it instantaneous, little hope for intervention that can be seen in overdose cases, for example. Yes, ownership makes it more likely for a successful suicide attempt. But one cannot simply control "for other factors such as mental illness", that is just not possible. Suicide comes as a result of mental distress, this may be a long term "mental illness" or a short-term situational distress, such as relationship breakdown, in short, it is nothing that anyone in the field "controls" for, because you can't do it.
In simple terms, if you are in a state of such mental distress as to attempt suicide, you are equally likely to make that attempt whether or not you have a gun, but you are, of course, more likely to be successful with it, but equally, in simple terms, I don't think many are more likely to attempt suicide just because they happen to own a gun