If you're looking for someone to defend the GOP keep looking. I'm no friend of their platform.

I'm not offering solutions or presenting either party as superior to the other on this issue, just pointing out issues that I perceive from my position.

As far as why the GOP does nothing: why would they? They're perfectly politically content to let the Democrats do all the work, since all they have to do is keep opposing it for votes. The portion of their voter base that are one issue gun voters need only silence and inaction to secure their vote.

Politically, the moment the GOP suggests anything that even suggests a hint of gun control, there's a riot in their base. So literally doing nothing but opposing Democrats on guns both attracts and holds those votes...it's the perfect political situation for them in that not only is a minimum of effort required, but the maximum benefit is realized specifically by saying the least about it.

As far as why no mental health proposals, that's easy too: that costs money. Any sort of initiative on that front means increasing taxes or pulling that funding from elsewhere, and if there's anything the modern GOP is clear about, it's that healthcare isn't their deal. Hell they're still adamant they want to take healthcare options away, why would they want to add more?

Ultimately it's just lazy and effective politics. By refusing to even talk about it, they can maintain the status quo, which is a situation in which their voters support their silence on the matter and opposition to proposals to upset the status quo...and on the other side it makes a huge issue for their politicians because while the GOP base wants no new gun control, the Dem base is screaming at them to do something. By preventing anything from being done, the GOP is making their opponents' lives that much harder too.

It's not that the GOP likes the shootings, it's just that being anti-gun control is such an easy position to maintain politically while securing a big chunk of votes.