My recent experience
If you can afford it, do both. As a retired healthy medical provider who became Medicare eligible, I had a PCP in one state who accepted Medicare. However I was moving to another state. Getting into a PCP as a new Medicare patient is difficult and can take months. I did find one in the new state, but ended up having to go concierge for some hormone therapy. So I was "all established" with 3 practices while I transitioned my move.
After avoiding COVID for almost 3 years, I finally got it in June while staying with my son and helping with my new grandson. I was very sick. Calling the PCP was impossible to get thru to a person. I used portal for both explaining I'm very sick, do I come in, should I go on Paxlovid, etc. Never got a response from first PCP office. I contacted the concierge last thinking that since their specialty was hormones, they wouldn't want to deal with COVID. My concierge practice is who called me right back and called in Paxlovid. The new PCP got back to me the following afternoon on the portal saying a script was called in and "per protocol" they were billing Medicare for a "telemedicine visit." That script never made it to the pharmacy and by the time I would have gotten thru again and waited, I would have been beyond the window of treatment.
If you can afford having a concierge provider, it cost very little to remain established with a PCP that is not concierge. This is in case your situation changes...such as you move, your provider dies, retires, or changes practice, and/or you want to see a certain subspecialty and only one can get you in quickly. If they have an objection to that, I say, I travel a lot so need to make sure I have access. If they still object, keep looking. You do have to keep them both informed of what the other is doing so you do get the best care.
As far as the comment about staying away from a provider that doesn't take insurance, I disagree. Our practice only took traditional Medicare. The first 3 years, we were in contract with several insurance companies but insurance companies do not play fair, so we dropped them all. We did submit claims and charged a fair price, but patients were responsible for the full fee. Medicare rules are very straight forward, only change once a year and you are given time to consider the new rules. Insurance companies deny claims to delay payments, misrepresent what they do and do not cover or make it difficult to find out until you file the claim, the practice has to negotiate a fee schedule every year, the list can go on and on. Insurance companies are in the business of making money; Medicare is trying to make best use of the money they are given.