She willingly goes to the castle probably to die in place of her father

And? How does that make it a problem?

Fathers are not supposed to let their children die for them. (Or Mothers)

My mum has The Uses of Enchantment, so I can't peak at what the psychoanalysis there was, but at least wikipedia lays out this pertinent plot point:

The next morning as the merchant is about to leave, he sees a rose garden and recalls that Belle had desired a rose. Upon picking the loveliest rose he finds, the merchant is confronted by a hideous 'Beast', which tells him that for taking his (the Beast's) most precious possession after accepting his hospitality, the merchant must die. The merchant begs to be set free, arguing that he had only picked the rose as a gift for his youngest daughter. The Beast agrees to let him give the rose to Belle, only if the merchant will return, or his daughter goes to the castle in his place.

The merchant is upset, but accepts this condition. The Beast sends him on his way, with jewels and fine clothes for his daughters, and stresses that Belle must come to the castle of her own accord. The merchant, upon arriving home, tries to hide the secret from Belle, but she pries it from him and willingly goes to the Beast's castle.

Fathers are not supposed to let their children die for them.

But that doesn't make the story problematic, it makes the daughter heroic. He didn't let her do anything. Give her some agency, nuh?

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.