Its quite obvious that she wasn't actually saying people with MH problems think they are more special or interesting than anyone else but making the point that, just because other people don't inappropriately share their MH issues, it doesn't mean they dont have them or that their mental wellbeing isn't less improtant.
She was referring to the people who are suggesting that because this woman is sharing hers, other people should get invovled. Only sharing doesn't make her more worthy of anyone's consideration than anyone else.
I've had Mental Health issues in the past (arguably I still do now but I manage them well).
At the point when I was most vulnerable, I would have been the person to respond to her because (I've since realised) whilst I was fighting someome else's battle and supporting them, I was, and felt justified in, neglecting my own needs with the excuse to myself that I didn't have time or that mine could wait because theirs were more important. Or it just provided a distraction to me. Whilst never asking anyone for support myself.
It's rarely people with strong boundaries who get involved in these situations and often those who have the least capacity to give who do.
Eg there have been countless threads on here over the years where people have posted because a friend and their MH is drainng them in terms of their own mental resources and health. With very few exceptions, the poster then goes on to reveal that they too have significant MH issues.
Many people who have had experience of supporting someone with significant MH issues also know that it isn't a case of "sparing a thought... every now and then." The lack of boundaries often means that the person who is struggling doesn't ever consider the needs or capacity of other people and just ask more and more and lean on them more and more.
Again, there was a thread earlier this year from someone in that position who just couldn't see that the friends who usually offered them most almost constant support weren't actually their 24/7 personal resource and resented the fact they were prioritising their own pre-existing plans rather than drop everything to speak to/see then when they needed them.
Even when someone isn't struggling themselves, that's an awful lot to ask of them.