First: Not true, you don't need a background check to buy a car. You do for a firearm, unless it is purchased in a private sale, a time when you couldn't enforce a check anyway.
Second: the closest thing to Gun rights for liberals is voting rights.
Restricting gun rights is not like licencing a privilege, it is like restricting voting rights.
If you can restrict gun rights without a constitutional amendment, then you can restrict voting rights without a constitutional amendment.
So a county could remove women's suffrage, or a city could remove people of color's right to vote.
Do you get the problem with that scenario?
Even the ACLU agrees with me: Until the No Fly List Is Fixed, It Shouldn’t Be Used to Restrict People’s Freedoms
There are restrictions even on free speech.
Once again no one is talking about removing rights. There are already restrictions and regulations in place for voting rights. There are regulations and restrictions in place for guns as well that many believe are inadequate and should be revisited.