As it is, the House is more representative of the population of the US than the Senate, so it's fair to say the biggest barrier to increased gun control right now is the quirky nature of the US political system, not popular sentiment.
That bill would do nothing to combat these mass shooters. I can't think of any offhand that have bought their weapons through private sale. Most don't have a criminal record so they just buy them from random gun shops.
So if it won't solve anything related to these incidents, what's the point?
No, it wouldn't solve everything. But if we wait to pass a law that solves everything, we'll never pass anything.
It's a common founding myth among gun rights people that "The American People" are pro-gun, but it's just not really true. It's a particular driving issue for a particular subset of the republican base, and beyond that opinions aren't as strong, but are broadly pro-gun-control.
It was a flawed list of too-specific rules that were self-contradictory or incomplete in a bunch of places, and indeed that became part of the mythology about it in gun circles.
But it also directly outlawed clones of the Kalashnikov and AR-15 rifles that have been preferentially used (for fairly obvious reasons) in much of the recent violence. It worked, within its domain.
No one likes the ACA either, but you can still buy insurance with a pre-existing condition. Same deal. The gun folks like to conflate "flawed law" with "useless law", but that's not how it works in practice.
