For my last 2 BasicMed physical exams, I ended up going back to my AME who issued my 3rd class medicals. My primary physician didn’t seem interested when I broached the subject. My AME guessed that something like 90% of primary care docs wouldn’t sign off on the BasicMed checklist due to fears of unknown liability. As an AME, he thinks he’s not liable for any issues that may get missed on the checklist since he doesn’t have a “clinical relationship” with the pilots (e.g. he’s not responsible for treatment or providing medical care).
I live near the border with Canada, so there is a plausible reason to keep getting 3rd class medical since Canada does not recognize BasicMed.