I just noticed that in the OP, pushing the wheelchair is described as a "privilege." Ms. Taylor and her companion may not view it that way. Mr. Nutrax and I do not. We discussed this thread and agreed that my pushing his chair is a kindness, in that I am performing a chore that is difficult for him to do, but it is not a sign of disrespect to allow someone else to perform the kindness. He does not expect me to insist that only I can do the job, not is he insulted if I decline and suggest that someone else do it.
In fact, if I were to insist that I was the only one who could properly push the chair, he would view me as controlling, not respectful.
As someone else mentioned, Ms. Taylor is wealthy. She is no doubt accustomed to, and comfortable with, paid staff who see to her physical well-being and comfort. She might feel that she would not be respectful of her companion to ask him to do this.
He also may have tried to push her and been remarkably unsuccessful, so they agreed he shouldn't do it any more. Pushing a chair is not easy. You need to learn special techniques for navigating things like curbs, thresholds, or railroad tracks. There are also special techniques for assisting people in & out of a chair. Perhaps he just doesn't have the knack.
In any case, it is their decision that matters, not anyone's speculation. As a Jane Austen hero said when the heroine worried about an engaged couple's relationship to each other: "Their hearts are open to each other, as neither heart can be to you; they know exactly what is required and what can be borne."