You need to understand two things. This is not a simple "Ban all guns and problem solved"
There are 2 issues at play here. How do you ban them, and what is your plan to get back 400 millions guns that are already in circulation currently?
You can introduce law to ban guns and make them illegal overnight, set aside constitution issue (Which I will discuss later) even if Senate passed the Background check bill or even ban Assault Firearms, then what? You only ban what people can get AFTER the date the bill is signed, as I said, there are already 400 millions firearms in the US both legal and illegal. You have to do something with those thing. Gun buy back only would work if you assume everyone is going to follow the law. If I am going to use my gun and commit crime later on, I would not possibly hand them over because I wouldn't care about the banning.
So the only way is for PD to look at records and go to everyone house and collect them, but then that's another 6th amendment right issue there, you can't go into people homes without either an invitation or a warrant. And accessing which people own what itself is a violation of 1st Amendment right (Government cannot look into privacy detail without a warrant) which mean you would either need a lot of warrant to go after gun owners. I am not sure SCOTUS are all for that, that's a lot of civil right violation there.
Then you have the 2A thing you need to go over, everything I said above is not going to happen unless you can somehow overturn 2A, which itself entrenched quite strong and there are no possibility on this is going to overturn, not with 6 Republican justice on SCOTUS. And then even if some how SCOTUS overturned 2A, you still need a bunch of Senator (This will probably fly pass the house) to agree on the bill, which mostly likely you will get is a filibuster.
Sadly, what I see from the school shooting is that gun sales are going to spike, because either people have a misconception of guns bring you protection (It's not true if you don't know how to use them responsibly) or they are afraid something is going down with gun law and stock it just in case, school shooting is good for firearms business, especially when those shooting is in Texas or Arizona or Florida.