Google prince Charles interference or similar and you get a huge number of relevant returns. You won't get this for any other member of the Royal Family in modern times, proof enough that something very different is going on.
I find it difficult to believe you think this is only speculation, given the clear evidence in bricks, stone and absence in some instances. Just a couple of examples. It was Charles who kicked up a huge fuss about a very sensible National Gallery extension, leading to it being changed into a compromise that suits neither him nor many others. It was Charles who created the Poundbury township which continues to this day extending over grade 1 agricultural land. Anywhere else and a threat of building on much poorer land is met with huge opposition and often failure, but at Poundbury Charles got his own way for a development which has clearly lost it's way from what I saw on my last visit recently.
But as has been clear from the outset, this thread is primarily about an undesirable precedent that can lead to royal influence in the governance of this country. The behaviour record of Charles was used solely as an indicator of what is possible. What we've had for approaching 300 years in the balance of Royalty and government has worked well and is in fact a valuable protector for the continuation of royalty. That should not be changed to suit the desire for publicity of an increasingly unpopular prime minister. Like all of his ilk, he seeks the reflected glory of fame from the company of others, be they a Queen at a cabinet meeting or a pop star at a party conference.
Why anyone would support that beggars belief.