Which psychologists are you talking about? I routinely refer my patients to social workers in the community who are excellent psychotherapists, and I have good working relationships with the social workers in my clinic. I work in a department where most of the other faculty are physicians, and I am thankful every day that I have my job and they have theirs. Admittedly, I was on the fence about med school versus graduate study long ago, but my experience working alongside physicians has validated my decision time and time again.
I suppose if you look hard enough you will find someone who matches the trope you are describing. I suppose I could have soured on the medical profession a long time ago as I have plenty of grist for that particular mill. But I find it more conducive to my well being and professional happiness to cultivate relationships with all kinds of professionals who care for patients, learn what I can from them, and nurture my identity as a psychologist.
You've gotten a lot of opinions here, most of which seem to be well intentioned even if they're not warm and fuzzy. We as psychologists look as these issues through the lens of our own ethics code, and to expect differently of us would be unfair. Why you would pursue a PsyD with no intention of becoming licensed to practice psychology is, quite frankly, beyond my comprehension, but I'm not here to examine your motives. If you are as transparent with your clients, your colleagues, and the general public as you are being here, then you are falling within the letter of the law, but you may still very well be looked upon as a [benign, somewhat confusing] outlier in the professional community.
My only concern is that, despite your transparency, clients who choose to take time out of their day and the money out of their pockets to see you may be vulnerable to a "bait-and-switch" situation once you disclose that the doctorate holder sitting across the room from them is not actually a licensed psychologist, unlike 95+ percent of the other PsyD holders who are in clinical practice. Some people won't care, others will, and some will cut their losses and enter treatment with you even though you're not the type of professional they expected. As others have suggested, it's really on you and your conscience as to whether you can be OK with that.