A student must be able, for example to clear a fully flooded mask repeatedly, correctly and fluidly. That's 1 item from the checklist (mask clear-fully flooded) plus the definition of mastery. Together that makes up one so called "performance requirement". In the PADI system there are couple of dozen or so "performance requirements" of this sort that comprise the standard for the open water course.
When put like that, how are you going to hold students to "a higher standard"? Is there a better definition of mastery than "repeatable, correct and fluid" when it comes to student training?
It seems to me that if you made the standard objective, instead of subjective, it would benefit everyone - AND then you could also understand what it would mean to hold students to a higher standard.
If you want a student to clear a fully flooded mask, you can say "they must clear a fully flooded mask repeatedly, correctly, and fluidly." Or, you could say "they must clear a fully flooded mask 3 times, while maintaining their starting depth within +/- 3 feet, not touching the bottom or breaking the surface, maintaining trim within 30 degrees of horizontal, and completing the skill in less than 30 seconds." That's just an example. How is it better for anyone to have "standards" that are totally subjective, instead of quantifying the standards?
With an objective standard defined, holding a student to a higher standard becomes pretty easy to understand. E.g. they have to maintain depth within +/- 1.5 feet. That would be a higher standard.
Note: Whether a particular agency allows an instructor to require meeting a higher standard is a separate issue that I am not invoking here. But, if an agency specifically desires to prevent instructors from adding requirements in order for a student to achieve certification, then it definitely seems that the PADI subjective way of defining standards (used as an example only because it's the one that has been cited in this thread) fails to achieve that. This thread already has several examples of how an instructor can bend the current standards to suit their desire to require the student to achieve more than what the standard actually requires.