I know this is pedantic but the subject is a quote, so your parent is correct. It really should have either used square brackets around the name, or been written differently. Personally, I feel that once you open a double-quote there's a certain responsibility.
The adjacent tweet from the CEO mentions Shay by name. I'm usually a stickler for this kind of thing as well, but this one reads as entirely unambiguous to me.
Our window for reaching its peak was gone. I sat in our tent, unable to do anything, while my dream of climbing Mount Everest faded
It's wrong to quote them as saying
"Our window for reaching Mount Everest's peak was gone"
Even though the "its" is clearly referring to Mount Everest. There are a couple options, the simplest being:
"Our window for reaching [Mount Everest's] peak was gone"
What worries me is that people don't know this. That you can just substitute/fill in a quote because it's obvious that's what the person meant. Rather, it should be thought of as a recording device that can only play back exactly what the person said.
