The Season 13 holiday special, Eve of the Daleks, included a scene where Yaz was openly declaring her attraction to The Doctor, and crying over the unrequitedness. My belief is that Doctor Who is being diminished by such talk.
Put myself in the shoes of traveling with someone like The Doctor, and I wonder how I'd feel about it? Would I be enthralled by the person?
That's an important question to ponder as we think about this piece in Eve of the Daleks that was telegraphed a long ways back. Is it ever appropriate for The Doctor to get hooked up with one of the Companions?
This episode contained the most explicitly presented ~L~O~V~E~ interest between The Doctor and one of his/her Companions. Namely, in an episode where everyone was killed several times (thanks to a timey wimey timey loopy thingy), Yaz had a moment where The Doctor had given some orders, then went charging off to do something, leaving Yaz extremely worried, and basically crying on Dan's shoulder.
The immediately previous episodes, the first of The Flux, we saw The Doctor and Yaz narrowly escaped death by falling into an ocean of bubbling acid. For the culmination of that sequence, they discovered where The TARDIS was hiding, fell into the front door of The TARDIS, where there was a net and a mattress conveniently present in the Console Room. Seeing that I let out an inner groan saying Please don't let that Mattress be a permanent fixture in the console room and Please don't have them doing hanky panky in the console room!!!
Thankfully that mattress was dispensed with almost immediately. But the implied ~F~E~E~L~I~N~G~S~ were there anyway.
I find this to be a further slide down a slippery slope we're unfortunately on with New Who. It's yet another reason that I'm exceedingly glad we will shortly see Chibnall exit the role of Doctor Who Show Runner. His ideas are simply NOT HELPING Doctor Who in any way. He needs to stop doing these stupid things.
The Sexualization of The Doctor and Companion relationship
Starting with the 2005 revival of Doctor Who, there's been various forms of unwarranted ~L~O~V~E~ vibes between The Doctor and The Companions. This started right off with Rose, and her flagrant sex-charged infatuation with both the 9th and 10th Doctor. The Rose story arc culminated with an actual ~L~O~V~E~ relationship with actual hanky panky in an alternate universe with a clone of The Doctor who was destined to have a human lifetime with no regeneration.
Then there was Martha Jones, who held onto the unrequited lover bit, a meeting with Sarah Jane Smith that turned into girls gabbing about their relationship with The Doctor, etc. With Matt Smith's Doctor, there was flagrantly implied stuff between him and Amy much to the chagrin of Rory. Later when the Ponds got trapped in Manhattan, Matt Smith's Doctor apparently fell head-over-heels for Clara, and later apologizing to Clara as the Capaldi Doctor, after a sequence of episodes where the two were essentially cheating on Clara's actual boyfriend who she should have married.
I was never happy with any of that. An especially egregious episode was the 2013 episode, Time of The Doctor, which was the story where the 13 Regenerations limit was lifted. It started with scenes in which The Doctor was naked, then put on some kind of holographic representation of clothing, but was still naked. Then he ordered Clara to also become naked, because they were heading to meet The Papal Mainframe, and for some reason that cult required nudity, leading a scene of unnecessary nudity references and The Doctor leering at Clara.
The Doctor is a nigh-on-immortal-demigod (as Professor River Song described him). What's he doing leering like a 13 year old at a naked woman?
Power dynamics between The Doctor and Companions
I'm not a boring old man who's screaming at kids walking on my lawn, and ordering women to dress decently. Nope, that's not me at all. I'm all for people having love interests WHERE IT IS APPROPRIATE. It is completely inappropriate for The Doctor to develop ~L~O~V~E~ interest with the Companions.
The issue is about power relationship entanglements.
For example, it's completely inappropriate for a Boss to get into a relationship with an Underling. It doesn't matter who it is, nor how honest and loving they are of each other, The Boss has power over The Underling. This gives The Boss an inappropriate power hold over The Underling which will poison the relationship.
The relationship between The Doctor and The Companion is similar enough to make it simply and plainly WRONG.
The Doctor is essentially immortal, the Companions are not
Why would The Doctor want to form a relationship with someone who's only going to last for 70 years at most. The Doctor will live hundreds of years beyond that person's life, yet will mourn and grieve over the loss when they do die of old age.
It's instead more appropriate for The Doctor to focus on relationships with similar beings. Other Time Lords, for example.
There was a discussion between Missy and Clara in an episode where the two wanted to find out how to locate The Doctor. Missy informed Clara that she was little more than a play-thing to The Doctor, and that Missy and The Doctor had a relationship spanning several thousand years. We do know that Missy/Master and The Doctor have been going at each other for a very long time, yes, of course. We also know that Missy was really trying to get a hook into Clara at that moment. But, Missy made a very fine point, that there's little reason for The Doctor to be properly ~I~N~T~E~R~E~S~T~E~D~ in someone with a short lifespan, like Clara.
The history of Relationships between The Doctor and Companions
In Old Who, this simply did not happen. In part that's because most of the old school TV programs did not sexualize their characters. That certainly was not the case for Old Who.
For example, when The 1st Doctor and Vicky (the striking young lady who replaced Susan) decided to travel together to Rome, leaving Barbara and Ian alone in a Roman Villa, there was no hint of any hanky-panky. And the "no hanky panky" even extended to Barbara and Ian. Similarly, The 7th Doctor and Ace had a few mannerisms, like The Doctor bonking Ace on the nose in a very familiar gesture, there seemed to be no hanky panky.
An exception was The 4th Doctor and Romana 2. In one story, they spent awhile in Paris, dashing around Paris like two love birds. Character-wise there is a huge difference between Romana and the typical Companions, in that Romana was a Time Lord. As people, however, Tom Baker and Lalla Ward (who played Romana 2), developed their own love relationship off-screen and even got married.
It would have been extremely appropriate for a ~L~O~V~E~ relationship to form between The Doctor and Romana. Both were Time Lords, would have had similar lifespans, and would have worked well as equals.
There is no way for The Doctor and any of his/her other Companions to have a relationship of equals.