FIrst, for many of these initiatives being proposed, we have Clinton-era data on how effective they were, and they weren’t. So, when the right says “that won’t matter”, they do so not on theoretical grounds, but often on empirical grounds.
Second, your “show me how much you care about mental health” proffer applies also to the armed-guards-in-schools notion. We have armed guards in banks, but not schools; can it not be reasonably concluded that we therefore care more about money than we do about children? Our politicians have armed guards; do we care more about our politicians than we do about children?
Large problems require out of the box thinking to solve. If gun regulations that pass Constitutional muster are insufficient, let’s come up with other ideas.