Try these, even if it has to be next time.
(1) see if you can snag the doctor's card, or possibly the clinic card if it lists several doctors. It often sits on the reception room, or ask the receptionist. It may list the direct numbers - not to the doctor, but to his receptionist, nurse, aid (not to some "call center", not to the "switchboard").
(2) try to go to places that have teams - where the doctor(s) and nurses and receptionist and schedulers are one team - at my huge clinic, if I use the client portal, the message goes to the team. Often, a team member can answer my issue/question/whatever. If not, they "nudge" the Doc. Maybe I just lucked out, but it works for me.
(3) If you end up having to make an appointment AND thus have to pay a co-charge, write a protest or appeal to the insurance company, explaining politely that they shouldn't have had to pay for an office visit, and "cc" the Doctor and the clinic. It isn't necessarily that the insurance company will pay, but they will ding the doctor, maybe his clinic administration. Maybe something will change.
If that "alienates" the Doc, fine. You didn't want him. Get another one and tell him why.
Stay safe.