The overarching problem with more gun laws, including universal background checks, is that nobody on either side of the issue will ever be able to prove to the other side that gun laws were/are responsible for decreasing (or increasing) the crime rate, even the gun crime rate. Sociology is not a simple formula where you can plug in gun laws to get predictions of changes in gun violence. Correlation is not causation. Culture drives crime and gun crime more than gun laws do. Etc.
Since there's no rational statistical argument that background checks impede criminals, and since you can't make a retrospective statistical argument about gun laws affecting crime rates, all that's left is the emotional tact: gun crime is bad, we have to do something, and universal background checks are one available option, and anyone who's legal shouldn't have a problem with it because they'll pass (eventually... though they might get delayed or even rejected until they call up NICS and get it sorted out).
Universal background checks amount to no provable effect on gun availability to criminals. They impose a burden on everyone who wants to transfer guns privately. They impose scheduling and transportation costs to meet at an FFL. They amount to more FFL paperwork, more gun ownership records for the ATF to illegally collect and archive, more NICS check data for the FBI to illegally archive, and more costly "private" firearms transfers, because FFL transfers are not free, even if NICS checks are.
If you want to reduce injury and death, get involved supporting vehicle/traffic safety improvements (real safety, not revenue generation), or improved cost/benefit preventive medical technology. Those kinds of efforts have far more likelihood of benefiting people.
If you want to reduce the risk of getting killed by someone with a firearm, consider carefully the kinds of people you associate with, and the kinds of people they associate with, because that has vastly more impact on your safety. Compare that to the chance that some crackhead or revolving-door ex-con who wants to hold up a store you happen to be shopping in, or picks your home for a hot burglary, is going to be deterred in his plan: because of universal background checks, he couldn't get a gun transfered through a FFL, so he gave up, and now you're facing a robber or a home invader with a knife or pepper spray instead of a gun. I don't think so, but that's me.