I find it hard to believe Meghan was ever told by anyone in power that she couldn't see a therapist. The person she (wrongly) approached in the palace was in the Human Resources dept which was for the servants. She was told by him that he could only make such arrangements for the paid employees. The staff for the royals, everyone from window cleaners to cooks and maids numbers the hundreds. Naturally that group of servants doesn't include the people they serve.
Diana had a regular therapist. We know Prince Phillip saw one at one time, and who knows who else, it would be confidential patient information. So I seriously doubt that Meghan would ever have been denied help if she had asked one of the other royals to arrange it for her. But I don't know why she would do that. All Meghan, herself, ever needed to do was call her regular doctor and ask for a referral. Then call for a driver, the same way she calls for a driver to take her to baby showers, pre-natal appointments or anywhere else. No need to call a Uber.
I feel sympathy for anyone with depression, but I don't understand why Meghan, a 39 year-old woman, couldn't have picked up her phone and made an appointment for herself. No one else even needed to know a thing about it.