Not exactly, but I'll get back to this at the end.
I can only find some stories about staff reductions from budget cuts around 2011/2012 time frame. I'm not aware of any more recent budget cuts. The up surge in gun sales during the Obama administration, and the rising popularity of suppressors has caused them a huge workload that they struggle to keep up with. Processing for NFA items (more on this in a minute) has been pretty slow the last few years, for example. I'm not aware of anything else going on at the moment. They've always had a hard time with record keeping for the purposes of traces (more on that below, too.)
They never actually have. It's not really one of their functions. More on this, too in a minute.
The answer is zero. The ATF's role isn't one that would have a very direct impact on preventing shootings.
With regards to private firearms sales, the ATF's role is primarily enforcing regulations. You have to have a license to sell firearms or manufacture firearms for sale. The ATF is the enforcer of these licenses. The National Firearms Act of 1934 made it more difficult to obtain certain classes of weapons, including fully automatic weapons, short barreled rifles, short barreled shotguns and silencers (among others). It did not make them illegal, but there is a requirement for a tax stamp to transfer an item on the NFA list. The cost of the stamp is $200 (fairly prohibitive in 1934; the average annual income was $1600 according to Google). The ATF manages the applications for tax stamps. Finally, when someone purchases a firearm, they fill out ATF Form 4473, which the ATF processes.
When you fill out a form 4473, you're basically attesting that you aren't a prohibited person. Lying on the form is a criminal offense. The background check is run trough the NICS by the gun shop after filling out the 4473. Most states also require a state and/or local level background check, and some impose further licensing or permitting requirements. So, from the buyer standpoint, here's what typically happens: You go to the retailer, you decide to buy a firearm, and the store fills out a 4473, then phones in the NICS check. If there's no state level background check or waiting period requirement (almost always is), then you get your gun and walk out the door. In Washington, for example, there's a requirement for a state background check for a handgun, which usually takes a couple of days to get back. It's waived if you have a valid concealed pistol license. In other states, there may be a mandatory waiting period.
Now, from the LE side, the ATF has a couple of additional roles it plays. First, it enforces laws against straw purchases. A straw purchase is basically when someone who isn't a prohibited person buys a firearm for someone who is. Most gun stores are actually very proactive in spotting these, and they're required to report them to the ATF. The ATF sometimes does sting operations to catch straw purchasers. Or, in the case of Operation Fast and Furious, facilitates the straw purchases (:facepalm:).
Another role the ATF plays is in tracing firearms. Their system for traces is frequently criticized (rightfully, IMO) for being antiquated. (I'd advise fact checking or bias checking anything from Vice, but that article is still pretty spot on as far as how bad the system is.) But as far as keeping track of who owns what? Not so much.
Edit: Another thing the ATF does to prevent trafficking is to investigate multiple gun sales within a short period of time. Link with info on that.
When you fill out a form 4473, the gun shop/seller keeps the form on file. All FFLs are required to keep the forms on file for (IIRC) 20 years. After that, they can do whatever they want. Many destroy the forms. If an FFL goes out of business, then their existing paperwork all goes to the ATF. This has been until very recently actual paper forms. (Some shops now use an electronic form, which should provide for better record keeping.)
I think the two links two paragraphs up pretty much cover traces. Traces are, of course, only useful after a crime has been committed. As for prevention? The ATF plays no role. That's on the background check system. The system needs an overhaul.
Here's a link from the ATF's What We Do page about their role with firearms.
In the case of many of the recent mass shooters, the systematic failure was in various agencies failing to get the shooter onto the prohibited persons list. We don't have nearly the level of reporting that we should.
Final thoughts: I get the impression that when you're mainly talking about the recent problems we've had here with mass shootings.I believe mass shootings are contagious in the same way that suicides are contagious. Remember the recent suicides of Anthony Bourdain and Kate Spade one after another? Calls to suicide prevention hotlines increased after these deaths, and there was a fair bit of additional advertising in my area for the hotlines. It's because it has been very well known for decades that when suicides get a lot of press, people commit more suicides. I believe the same kind of contagion is true in the case of mass shooters. Most of them who leave manifestos specifically mention their predecessors. Washington Post link. The Atlantic link. Scientific American link.
I think one thing we can do is figure out a way to get more agencies to report possible prohibited persons.
I think we have a fair idea of some indicators of potential shooters. The Parkland turd gave more than ample warning signs, but law enforcement failed at early intervention.
I think that once a mass shooting is in progress, it has been proven that rapid, competent engagement is the best response. That doesn't mean a school resource officer and two local PD sitting outside the building like cowards. It means competent, brave individuals meeting force with force. Like this man. (Also at Parkland, but off duty.) Or this one: Lone resource officer's quick action stopped the Maryland school shooter within seconds. Unfortunately, we don't have a sufficient supply of such people.
Disclaimer: I made a bunch of assertions in this post that I felt needed some links to provide further clarity/information. Aside from straight informational links from government sources, I'm not endorsing any of these articles. I scanned them and believe them to be generally accurate in their factual assertions, but it's up to the reader to detect any political bias or slant in the articles and to critically assess any large claims made in anything they read.
Well, this got quite long, apologies to anyone who suffered all the way through it.