I don't think the background checks are the biggest issue in the president's gun control gambit. Those will mildly discourage buyers but mostly add a hurdle to jump through. Folks who wouldn't pass a background check will find other ways to get their weapons, while law-abiding folks who want or need a gun will have fewer choices.
A background check for casual sales would mandate a two-fold process, where buyer and seller first agree upon the sale, then the background check is run. If it takes a day or two for the check, that will mean that the parties have to come back together after the check is run, so that a deal made at a Saturday gun show would have to be completed on Tuesday or Wednesday without the benefit of a neutral site.
That would take some of the steam out of weekend gun shows unless some sort of automated system that worked weekends could be implemented. Having some sort of sheriff's department kiosk where a deputy checks the database and gives folks a thumbs up in a few minutes would not be burdensome, but making a second trip in (possibly to a more distant site) to get the gun would put a damper on the shows.
However. the biggest change is the concept that anyone who sells a gun needs to be a federally-registered gun dealer.
The measure clarifies that individuals "in the business of selling firearms" register as licensed gun dealers, effectively narrowing the so-called "gun show loophole," which exempts most small sellers from keeping formal sales records.
That potentially covers any sale of a handgun, even to a registered gun dealer. That will effectively dry up any secondary market for guns, as folks would tend to either hang on to their weapons or give them to authorities if they have to get a license to get rid of their guns.
We now see the prospect of a lot of weapons lying unused and forgotten, or sold under the table anyways. Having to run a background check on a potential buyer is a minor hurdle, but making the seller get a license to sell issued at the government's discretion and time-table chills that market rather quickly.
That will dry up a secondary market for guns and take a lot of used guns off the market. That's a feature if you're a gun control maven but a bug if you're in the market for an inexpensive gun. There will be fewer used guns for sale and they will likely need to be sold by professional gun dealers, since the casual seller will get driven from the market.
An interesting side effect here will be that it will be poorer folks who will be impacted by this change. They're often in tougher neighborhoods and more in the need of weapons for self-defense. Taking away that option will see them tend to leave that area faster, while your gentry, who can afford to go and buy a new gun as well as afford to fix up the fleeing poor person's place, move in.
Making guns more expensive will price some casual buyers out of the market, putting something of a damper on the gun culture that often pops up in downscale neighborhoods. The gun-control fan would point out that having that self-defense heater can turn into a self-offense weapon if severely offended. If self-offense is more likely than self-defense, you can make a case that taking used guns off the market would be a good thing.
However, that tends to makes self-defense a luxury item, affordable only by the well-off, while the law-abiding poor are fair game. That's not something would mesh with the left's defense of the poor, unless they think the poor can't defend themselves and need supervision by the nobility.