Lack of justification for operational definition?
That's pretty subjective and could be leveled against almost anything behavioral or psychological.
The thing about the Big Five is that it has surfaced in all sorts of contexts. You can maybe say maybe the 5 per se is not well justified (as opposed to 4 or 6 or 7, for example), but if you take enough ratings of a person, some variant of those 5 will probably work as reasonable summaries of the ratings, and they will account for a substantial chunk of their predictive variance. The thing too, is that if you take other types of variables, like clinical symptom ratings, or diagnoses, you start to see roughly similar types of attributes become prominent.
The Big Five is a descriptive model of how people perceive others. There's a lot of evidence for certain mechanistic processes being heavily involved in some (e.g., positive emotion in extraversion, negative emotion in neuroticism, behavioral control with conscientiousness, etc.) but I'm not sure the original idea behind the Big Five was mechanistic -- it was a hypothesis about major dimensions that could summarize social perceptual data. It's like classification in biology pre-DNA era. People have some ideas of how things go together, and find it useful for organizing descriptions and measuremnts.
It's like if you did unsupervised DL modeling of all the videos involving humans you can find on the web, and found that their classification could be accounted for by 5 major vectors, almost all the time, regardless of sampling. Wouldn't you want to know that?
Many other measures are very mechanistically well-justified but lack ecological validity, in the sense that they are very narrow predictively and not well outside of laboratory contexts. That's fine, there's a tension there between predictive bandwidth and depth, but if you want any kind of rating of a human being's behavior and experience, you enter at your own risk if you think you'll measure something that's radically different from the Big Five (or something subsumed by cognitive measures). Can you do it? Sure, but a lot of the time no (see: grit).