There is absolutely nothing to prevent it from happening by having a background check.
Tread carefully with that word "absolutely", there are certainly things that prevent or hinder law abiding citizens from owning a firearm due to background checks. Identity theft, similar/same name as a felon, etc. Something that prevents or hinders...sounds awfully similar to restriction.
More broadly, if you say background checks are not a restriction. What is the purpose of them? Is the purpose not to restrict some people from owning firearms?