Right which is how I remember it. She willingly goes, but he lets her - both thinking she is going to die. He should insist on going. Also how does she pry it out of him if he is really willing to let her pry it out. He is sacrificing his daughter. She is a willing sacrifice, but he is willing to let her die for him. Not his fault that is not how it works out.

He should have agency too. He could insist on going.

Well, then there'd be no story. No Beauty and the Beast story, anyway.

It seems there is that problematic aspect to some incarnations of the story, but it's not inherent, and it wasn't in the original, nor the first televised version--am I correct?

We might be talking at cross-purposes a bit? You're talking more about specifics of the fairy tale and its adaptations, and I'm talking more about the generalities -- the influence of the trope on other work of fiction as well as on the way real women perceive their relationships with certain men.

Is your issue that you're a sucker for the versions where he is violent and coercive and harms or restricts her in some way?

I haven't seen an adaptation of B&B where the Beast is really violent or brutal, so no. Not so much. That said, my absolute favourite version of the fairy tale is the Jean Cocteau version from the 40's, which is beautiful but eerie and incredibly unnerving -- there is more menace and element of horror in that movie than any other versions I've seen. The Beast is very mannered and brooding, and remains a lot less cuddly compared to subsequent versions. When he is turned back into the prince, it's a crashing disappointment -- you want the Beast back. In a way, I think the problematic (or at least the parts I find problematic) aspect is inextricably tied to the appeal of the story. That doesn't mean it shouldn't be examined. As you can see, I have a lot of mixed feelings about it.

Oh, I just thought of another example. A couple of weeks ago, I got inadvertently sucked into mainlining a 30+ chapter fanfic about Sansa Stark/Sandor Cregane (from GRRM's Song of Ice and Fire series), which I think of as a quintessential Beauty and Beast pairing. It's a lot more obviously sketchy than the fairy tale version -- Sansa is like 12 years old when they meet and Sandor Cregane is violent sociopath with a scarred face. The fic was all about how she was the only one he held dear and had the power to transform him (through sex, I mean love, I would imagine), yada yada. While I was reading it, I enjoyed the story, then afterward, I was kinda creeped out. I guess you could say it was an id-fic I was sort of ashamed to have enjoyed. Bottom line is, It's a similar trope, pushed more to the extreme.

He could insist on going.

He was there. He left. That is his action. No one is taking away agency from him.

But you're talking about what she should be "let" do, and that's absolutely making it someone else's story. If it was about her father, his name would be in the title.

I could see Belle convincing her dad that he needed to stay and provide for his other daughters, rather than leaving all three of them fatherless and poor. My takeaway wasn't that Dad passed the buck, but that Belle was awesome.

the influence of the trope on other work of fiction as well as on the way real women perceive their relationships with certain men.

I think the issue is that the trope, to me, is "beauty is more than skin deep" and to you it's "you can love the bad out of him" or "you should love him because he's bad", neither of which I can tie back to this story. I'm sure there's some other that says it, but I'm not sure why this one's generalities are to blame when that's not the moral of the original to me--it seems to be something some adaptations introduced, but isn't inherent, since the TV show didn't need it.

But he is literally transformed because she loves him, so there's that subtext, that a good woman's love can change a man.

Now I want there to be a show based on The Frog Prince, where the curse is cured by the chick killing him.

He's transformed physically, but he's still the same dude. His personality and nature doesn't change - that transformation already happened when he was cursed. The physical transformation always struck me as shallow and too shiny, but helpfully avoided the bestiality part of their romantic relationship.

I'm also now thinking of poor Priestley from 10 Inch Hero.

What if it were translated to magical gender-swapping?