No true. In some states, like TX, there is no background check if you have a LTC or CHL.
Interesting. Learn something new every day I guess. In Oregon you go through the background check but if you have a CHL you start very very near the front of the line. When I was making my last purchase the guy in the line next to me did not have a CHL. He was told he was 4700th or something like that and they'd call him in a couple weeks to come get his gun. I went straight to 2nd in line and was out the door with my new gun .. even with Sportsman's Warehouse's very onerous paperwork check, double-check, triple-check, then manager-check process .. in about 40 minutes. I don't like waiting 40 minutes but there's worse places to do it than at a gun counter and, truly, though infrequent I've been in line longer than that to check out with groceries.
I can't see any reason anyone who could get a CHL would not do so. A buddy from work got his a couple weeks ago. I told him it would probably take about 45 days, that's what it took me 20-odd years ago. Heck no. The delay was only a couple days for processing before handing him his shiny new card. .. and that's Oregon, among the bluest of blue states. Why anyone from a more conservative state wouldn't have a CHL is beyond my understanding.
Tom