I don't think these two issues have much to do with each other, the current public conversation as you derisively describe it, notwithstanding. The availability and efficacy of mental health care across the country is one issue. Whether or not any change in public policy. access, surveillance, etc would alter the inclination of certain young white males to engage in indiscriminate public slaughter is quite another, and the two should not be conflated.
Regarding the former, I have no expertise, although for many years I had a close personal connection with someone who did. What I do know is that if you present to my hospital this morning with a heart attack you will receive prompt (the rule is 30 minutes), definitive therapy from a team of highly qualified experts who are literally standing by for such emergencies. If one the other hand, you present in the midst of a psychotic break, suicidal ideation, or any other psychiatric condition that warrants immediate acute care hospitalization you will likely wait, perhaps strapped to a gurney, in the "psych" section of the ER, for anywhere up to 96 hours until a bed can be found for you. If you are without means, you may wait even longer, given the dramatic decline in public facilities for the inpatient treatment of the mentally ill. On the outpatient side, the state of affairs is are equally problematic, in terms of both availability and efficacy.
But I doubt, I repeat, that this state of affairs no matter what one thinks of it, has much to do with the nation's failure to attenuate gun violence. Gun availability surely is part of the equation. But I don't believe that's the whole story. Something deeper and harder to define within the American psyche is also at work.
As far as your neighbors' resentment of public funding for children with learning disabilities, the obvious query comes to mind. What do you think their attitude would be if they blessed with, and loved, such a child?