Because he wasn't and it was an infringement as well as deception.
First, you have to realize the "at gunshows" part is misleading. All dealers at gunshows do perform background checks. It's sales between private citizens "at gunshows" that are in question. It really doesn't matter if the transaction came about because of an ad in the paper, word-of-mouth, or referrals from friends. -- it's a private transaction between two private citizens. So that part about selling to criminals is a myth. While Private sales do take place at gun shows the percentage of criminals attempting to gain weapons this way is miniscule.
Next argument is that in order to implement full background checks between private citizens, you must register all firearms (otherwise, how can you verify that a background check is being implemented). Registration is the first step towards confiscation (see what's happening in New York, California right now -- and North Caroline and Louisiana during hurricanes).
Last argument is that, even if it were passed, there's no process in place for the FBI to extend background check processing to private citizens. They can barely handle the load right now from gun dealers. Which we even have Biden admitting the Fed doesn't have time to check and prosecute.
So -- by passing such laws, you will do something that makes people "feel good", but will have no affect on criminal activity or prevention of future crimes. Basically, you will have ignored the problem.
The bottom line argument is that if you feel action is necessary to reduce or eliminate criminal activity, then you're not accomplishing anything by implementing laws that will not reduce or eliminate criminal activity.