The Rogue storyline proves that New Who has a writing problem that has nothing to do with being too woke, and, in fact, at least in parts more with being not woke enough. What reason could there be to hint at, introduce, develop, carry out, end, and bury a love interest all within one single episode, if not as concession for people against those story line to be able to avoid it altogether? Disney isn't actually progressive, they never were, and where corporate pride/POC representation is implemented it's because the cost calculation tells them it pays off, not because they actually represent any of those values.
The Doctor would never give up on anyone. Heck, in the next episode he cried more about some random guard post than the man who proposed to him. Until he at least attempts to get Rogue back I will have to assume the Doctor has either been replaced by some kind of doppelganger, or the writers at Disney lack the balls to actually portray a modern and gay Doctor after all.
I have a cat just like yours with the same problem and can attest that it really does help. My fluffy guy really was no fan of the process, but he enjoyed summer much more afterwards. There really was an immediate change in his attitude, less apathic, more playful, cuddlier, and just happier overall. Will certainly do it again this year.