I think linking the two is making each a bit of a smokescreen for the other.
Both are completely separate, but equally important issues.
It is one thing to talk about how mental health care will nip mass shootings in the bud, but mental health care is needed for far more than just mass shootings. Far more people die from situations related to, or resulting from mental health problems than what happens during a mass shooting. These people need help, not because they might get a gun and shoot up a mall, but because they or others might be hurt regardless.
As for gun control, just like how the mentally unsound need help for their problems, most gun fatalities come from unfortunate accidents at home, like children shooting themselves of each other. Those are the situation gun control is hoping to avoid, not the mass shootings but all the other accidents and killings. You dont need to be mentally unsound to have a situation arise where your firearm kills you, or a member of your family.
Both issues are important. Both issues need to be addressed, and both issues should be address separately. Mass shootings shouldn't be the catalyst for these discussions, we should be trying to improve both all the time for all the other reasons. Mass shootings are just shockingly newsworthy and so get everyone into a media frenzy.