Actually, I think many people LIKE the word "patient".
Whatever its derivations may be, I doubt that many people think of it as meaning "sufferer". I think for many people it means the role of the person seeking (and hopefully obtaining) advice and/or treatment from a doctor or other healthcare provider. (I shall use "doctor" for the rest of this comment, as I am a doctor; I expect that most, perhaps all, of what I say will apply to people in other healthcare provider roles.)
The role of a doctor is that of a professional. The duty of a doctor is not generally considered to be purely to provide what somebody wants, as long as they can pay the fee negotiated. Ethical principles apply.
The word "customer", however, implies that the patient can have whatever they want from the doctor, if it can be provided and a fee agreed, whatever harms the "treatment" they purchase may cause.
Or maybe you do feel that if you decide you want a treatment which, in a doctor's eyes, will cost you money, do you harm, and not benefit you, you should be able to demand that the doctor provides it for you because, after all, you're paying the bill?
Other terms are used in some circles. "Client" is commonly used in some specialities; and in other circles people whom I would call patients are called "users" - short for "service-users", meaning that they use the services of a healthcare provider (a term which seems to me to have overtones of "drug user" with the stigma this carries).
I wonder if you are mistaking the issue here? Do you perceive illness as stigmatising per se? If so, any expression used to refer to somebody who is or may be unwell could be or become stigmatising by association? If that's the case then whatever word you substitute for "patient" will, over time, acquire the same stigma.
No, I like the term "patient". I have no objection, when I am on the other side of the desk or on the couch or operating table, to being called a "patient". This is, as far as I'm concerned, a dignified and adult way to refer to me when in this role.