Why?

For one, that's a hell of a responsibility to give others, even well meaning others. I wouldn't want my friends to feel they had to speak on my behalf, nor would I do so on theirs. I'd say that's a healthy level of respect to have in a friendship.

But to a larger point: Why are you so concerned about conversations others may or may not be having about you?

Just to rephrase the point you just made: You're so worried that people who don't know you well enough to be sure of your preferred pronouns may be misgendering you with non-gendered pronouns, that you hope others will correct them on your behalf.

This is, at minimum, overthinking, and at worst a deeply unhealthy mindset to have.

The kindest thing I can say is that whether or not you consider this a problem, it's at least a rare one; nobody talks about you as much as you think. People's favourite topics of discussion are themselves, and the odds are this doesn't happen even a fraction as much as you think.

I have commented elsewhere already, but I think it's good to correct people when you know someone's pronouns are different to what was just used.

I am cis, but I am in community with many trans people, and am pretty good at remembering and using the right pronouns for folks. However, I wasn't always like that — I only found my queer community once I got to university, and I remember that I would often slip up when talking about an nb friend (who was actually the first person I met who used they/them pronouns).

I only got good at using the correct pronouns because of the support of friends who would firmly but non-judgmentally correct me whenever I messed up (and explain to me that I shouldn't make a big deal apologising if I did make a mistake).

My friends doing this helped me to get good at this much sooner, as well as shielding my NB friend from the emotional stress of correcting me; They once snapped at me when I slipped up, and later apologised and said that whilst they felt they were reasonable to correct me, that the anger they expressed was not deserved, and that it was basically overflow anger from having to deal with university bureaucrats and ignorant students who were not even trying to get it right. My point here is that I was a good faith person who was genuinely trying to be better, but my mistakes that occurred in front of my nb friend risked triggering a lot of stressy-upsetty feelings relating to being intentionally or callously misgendered).