I have been asked to do a presentation on my work during this review year. It is not part of my job description or part of my job, it is however one of my manager's team goals set for her by her team manager. Like you, I simply cannot do a presentation, nor do I have any desire to learn (late 40s, been there done that in other jobs, absolutely not worth the stress for me, and I acknowledge that fact). I've actively avoided jobs where I would have to do this.
The other side of this is I'm a valued employee, been there a long time, and in the right situation, I make good verbal contributions in other settings (I can do it in formal meetings with seniors, team meetings) and I am generally acknowledged as a careful, tactful and intelligent communicator at work. It's just that being on a podium so to speak, in front of others, - well I melt into sweat and shakes. I don't have any specific mental health issues as such, but I have been on anxiety meds for 15 years as I have had panic attacks before.
I've already raised this with my manager and she has suggested ways to make it 'work' for me - presently these ideas work for her a lot more than they do for me though. So this is still work in progress I'm afraid, but hopefully a compromise can be found. Also, I will die on this hill - they can try and make me do it, but it isn't in their interests, or mine - so why would they? It doesn't benefit anyone. And like I said, it's not a requirement of my job.
I realise this is a bit different to your job, but I wanted to offer some support in that how you are feeling is not uncommon at all. I hope your manager is helpful, I think going with 'suggestions' should be welcomed - and they need to accommodate these where they can. Good luck.