Regardless of what personal values you attach to the word, the game is very specific about the duties of a mentor, and that is literally ONLY tied to helping and mentoring sprouts, and not in any way tied to behaviour/ability once you take the symbol off, or play with other veterans. It says so in the description of the job, and again in the list you need to accept when you agree to become a mentor. If you ask the NPC, he explains it as well. Mentoring is specifically tied to Sprouts, not all players in general.
Personally, I totally agree and am very annoyed when I for example get a bad player in my Zurvan EX farm party who is wearing a mentor sign or something like that, because I also attach more value to the tag than the game itsself does, but that's just my personal thoughts on it and the standard I set for myself (which is why I specifically take off the crown when I go for content that I am new or inexperienced in) and not something that I think needs to be enforced.
As for symbols - as said at the beginning of this post, it's false information that everyone can wear any symbol. You can only wear the generic crown and the symbol for the requirements you actually completed, not any symbol regardless of what you can do or not

For example, while the crafting mentor crown shows up in my list of symbols, when I try to equip it, the game tells me I am unable to because I'm not a crafting mentor.
I agree with that, and assume the system will be adjusted. When they extended the free trial, they also extended the length of the sprout status to completing basic HW, so I assume it's gonna move up to completing basic SB once we hit 5.0.
But it's still gonna exclude endgame content, so I am still convinced that is not a proper requirement for becoming a mentor, as it is not needed in guiding sprouts who will lose the sprout status before they get into endgame :x