ToddStL

TL;DR: Is there a difference between embracing AAVE in the workplace internally vs. when your business has employees who regularly engage with the public?

As a trainer of Customer Service Representatives (meaning employees who interact with the public on a daily basis) I have worked for companies that embraced AAVE and those who didn't.
There are several reasons I'm listing "mixed" as my take on this. On one hand, because we are not talking about an office setting (where interactions are between people who have the opportunity to "get to know each other") I do feel that companies should be allowed to expect a level of "standard" English proficiency for this sort of job.
That being said, I've also worked positions where the clientele being served are predominantly minority. In these instances, the general rule is that AAVE is perfectly fine to use on calls.. I also speak passable Spanish. As a "stereotypical" white person, I have often found myself in a position (both while training and while conducting quality assessments) where I wasn't sure if a word or phrase should be considered "unprofessional" when spoken in a dialect or (what I thought my be slang) in another language. Rather than just marking people down for it, I've reached out to peers of other races and backgrounds, and have learned a lot (and am better for it.)
So here's the "against" argument I'd like to make: I 100% understand the argument as to why employees shouldn't be made to code-switch; however, since AAVE can be seen as "incorrect" by others (such as myself) how are we (I guess by "we" I mean white people?) to know if the person speaking AAVE is speaking AAVE correctly?
I'm not being glib. AAVE has its own grammar and rules (unfortunately, most of which are used to stereotype/disparage others) but (I hope this makes sense?) I would assume that there are speakers of AAVE who speak the dialect "incorrectly" in a way that others who are native to it would notice??? (or am I just flat wrong here?)
Also, am I being ignorant in suggesting that it's "ok" for a company to expect an employee with a "public facing" job to act/speak in a "commonly accepted" way? An example I'm thinking of is the ex NFL player, Aquib Talib, who had a brief stint as a broadcaster and spoke heavily accented (to my ear) AAVE. He no longer has this job (this could be a poor example because I think he might have some legal troubles as well?) but I'm curious if others think it was "ok" for him to be let go because of his dialect?

While I generally agree with your argument (which is much the same as the "against" part of my "mixed" answer) that in certain situations, it is most effective/preferred to speak in the most commonly accepted manner, there is also a certain "I still understand what they meant" aspect to be considered.
For example (and no, I am NOT "trolling" you; I simply feel it's a good example) you said "through a person through the loop" and the idiom is actually "throw a person FOR A loop." Also, you used the word "your" when you clearly meant "you're" as well as saying "you as a doctor is not trained enough" when (grammatically) it would be "you, as a doctor, ARE not trained..."