The goal of any organization that focuses on a strong culture is that their employees bring their authentic self to work. It makes perfect sense and provides a wealth of benefits:
Increased creativity and innovation.
Improved employee engagement and retention.
Enhanced collaboration and teamwork.
Increased trust and transparency.
Better understanding of customers and stakeholders.
Any organization that truly values its employees should strive for the idea of bringing your authentic self to work in order to accrue all these benefits and create a great workplace for their people.
I also believe a strong case can be made that a culture should want employees to bring most of their authentic self to work, but not necessarily all. This makes sense from an employee and employer point of view.
The biggest reason is that many of your employees, especially more experienced employees who are in leadership positions, are not ready to handle it. There are still too many "unwritten rules" that people live by that would end up hurting the culture.
I tell the story of one person I wanted to hire. Smart, experienced, exactly what we needed. Yet one VP kept hinting that I should think twice before making an offer. When I pushed for a reason, he finally said that the candidate had long hair and he wasn't sure that fit our conservative environment. To be clear, this was 2016 - not 1970. Imagine the response to tattoos or piercings!
Note: I too gruffly dismissed the VP (and may have hinted that he lacked a certain intelligence) and hired the candidate - who was awesome.
I have daughters who have exposed me to many things I would have never even realized existed before and even I might do a double take based on appearance or lifestyle choices. I take no pride in that, but I'm being honest.
If I told an employee to be authentic and they came in with a nose ring and multi-colored hair and I made a comment or questioned their decision, the damage to culture and trust would be huge. Not only for that employee, but for the entire team.
In the end, managers have to be honest with themselves and with their employees regarding authenticity. Otherwise, a well-intentioned idea will do more damage than if you never implemented that concept at all.