This ^ is toxic positivity. I am not unpaid therapist for my work colleagues. I have my own problems.
There’s no need to get into details or cross over into therapy territory, but a “hey, I’m currently not doing perfectly fine” is a much appreciated heads up for me. Showing that you do care to a certain extend helps people share those feelings.
Unless you don’t actually care and can’t tell them “get your shit together on someone else’s time, dude” because that would be a micro-aggression since they don’t like being called dude.
If someone is struggling, that means they're vulnerable. Forcing them to take a gamble by enlarging the "surface area" of the risk that they're exposed to (by advertising it to someone else) is shitty. It punishes people who can't take that risk (in other words, the people most in need of relief) and also (separately, possibly doubly) punishes the people who manage to successfully cover their own bases without slipping or otherwise indicating that any of this is coming at great cost.
Corollary: you should assume that anyone could be struggling at any time (or all the time), and default to giving everyone whatever latitude they've earned and deserve, whether they've had an intimate heart-to-heart with you beforehand or not.
I agree that it comes with vulnerability and risk. However, I believe that the way out here is reducing the risk, by building team structures where people can talk about these issues without being punished for it. That is: Teams where people are not expected to be robots and interchangeable, but humans with strength, weaknesses and temporary or permanent challenges that need addressing.
People are very much paid to interact with colleagues in constructive manners, including understanding and addressing concerns.