My dislike of BFE degrees is that it is too specialist too early, and too narrow.
Anielson asks the perfectly reasonable question about why quantum mechanics is somehow 'better' than pricing options.
The problem is that a sustainable career in finance requires that you have a broad skill set. If you only do the math techniques that can be done at undergrad level to deal with undergrad level financial modelling, pricing etc there will be chunks of maths that you don't do.
You might say that (for instance) number theory isn't big in quant work. However, one very well known quant stated quite clearly how this caused him issues once he started having to deal with the more advanced forms of random number generators used in modern finance.
That wasn't easily predictable, and I'm not going to even try and guess which flavours of maths will grow in value, but I'm completely certain that you will find yourself wishing you'd done more maths thant you had. That's independent of how much maths you have, the variable is how frequently you will wish it, and how much it will cost you.
"Modelling maths" which is not well defined, but I shall lump together as anything that predicts what physical systems will do, is critical to being a successful quant. The skill of being able to use the right methods, and having a good vocabulary of methods to choose from is a key predictor of you success.
That's why so many engineers and physicists go into this line of work, relatively few economists...
I reckon that I could take almost any 18 year old who has studied maths and physics and get him to manipulate Black Scholes, and do stuff like implied vols. By that I do not demonstrate my own value as a teacher, but the lack of value in that skill. To make money over the long term, you have to do something that most others can't. BS is not there any more.
You would be right to detect prejudice in my position on BFEs here, but remember what I do for a living. Part of that is reflecting the prejudices of the people I want to sell you to.
I guess I could have written that last sentence in a way that made me sound like a nicer person, but I see my role as calling it like I see it.
I don't believe a BFE is a fatal error, in fact I don't really believe in fatal career errors else I wouldn't have had one

I do believe in lost opportunities and realising you've screwed up as a way of dealing with where you are not where you want to be.
A BFE can be done well, at least compared to other BFEs, but I will share that no hiring manager has ever expressed to me that he would like to see candidates with a BFE, Not one. Ever.