I'm interested by this thread op. Especially that clip from Twitter with Gary Glitter which, aside from the horrific racism, certainly made me raise an eyebrow or three.
Gyles Brandreth is undoubtedly intelligent, well read and hard-working with bundles of energy . But to answer your question op, I think he is all over our screens atm because, I could be wrong, but he appears to be an unofficial spokesperson for Camilla. According to a puff piece he wrote about her for the Fail, he has known her for years;
"I first met Camilla when we were both teenagers in the 1960s. I was at a boarding school in Hampshire and visited her grandparents at their rather grand manor house, Hall Place, nearby."
They have since worked on various literary and poetry projects together.
I think it's interesting therefore that when you Google "Gyles Brandreth and Meghan Markle" a whole array of very unflattering articles by GB about MM pop up. Together with a series of articles about how kind the RF were to MM and how she apparently rejected that kindness in various ways.
Funny also, given this context, how he wouldn't exactly be pinned down about Jeremy Clarkson's recent hideous comments about MM although he did call it vile I think (I didn't see the TV programme concerned, I only read about it later).
And I heard him a couple of times (quite gratuitously imho) drop the sentence "Diana was an hysteric you know, it was quite obvious when you met her" in to two separate TV appearances a month or so back, which I thought was pretty shoddy given that she is dead, he has no medical training, and it's pretty misogynistic to imply that a woman is permanently "hysterical" in this day and age.
I think there is definitely room for a decent journalist to be doing a little delving in to how the Royal Family use "friends and supporters" to feed stories in to the press to "influence" the public's view of a person or situation and to investigate who those people really are.