Not only is there a demand for combined treatment, there is also some demand for physician-delivered psychotherapy. Some patients just see me for therapy alone, either because they don't want/need meds or because they have another psychiatrist prescribing meds.
I've had patients who already have a therapist switch over to seeing me for therapy as well even though it was basically free for them to see a therapist through their benefits. They felt they got more from me than their existing therapists. Fact is, most therapists out there are shockingly bad. They may be fine for problems of living but a lot are not very adept at dealing with anything else. They want to EMDR everything to death.
Round my way, a typical therapist might describe themselves as a "queer non-binary kink-affirming, sex-positive, BIPOC-allied, neurospicy, decolonized, trauma-informed, attachment-based EMDR/brainspotting/IFS therapist". I'm sure that appeals to some people, but it's a massive turn off to a lot of patients.
That said, providing combined treatment would not separate you from other private pay psychiatrists since most private pay shrinks offer this. One of the main reasons people forgo accepting insurance is because it doesn't reimburse adequately for psychotherapy, whereas you can make as much or more with insurance if you focus on pharmacotherapy.