we need publicly funded mental health care as much as we need gun control.
While the US definitely needs publicly funded (mental) health care, it will not address the gun issue. It doesn’t matter if a country has public health care or not, what matters for gun related deaths is either a) number of (civil) guns or b) (civil) war.
Do not give into gun nuts in this regard. Do not agree that the US needs both. The US needs exactly and only one thing when it comes to gun deaths: Fewer guns.