I wouldn't, for a few reasons.
This biggest is I have a number of friends who that sort of money would make important improvements in their lives, so why would I give to one rather than another? I'd be a bit worried it could open a floodgate and I'd end up being constantly asked and feeling "Well I gave to OP, it's only fair I give to them."
The second one is that private medicine does vary, from ones that will give you what parent wants, through ones that will keep leading you on to "you now need this next equipment/diagnoses/assistance", through to the excellent ones.
Unless I'd had first hand knowledge I don't know which one you've chosen, because the former two are totally plausible at a superficial level-they have to be to get the money in.
Throw in the money making scheme and I'd also be worried having donated once, I was going to find it was always "just need another couple of grand, and then we'll be fine."
Also in my experience of people I have known well, and certainly not saying this is you, OP, most of the people who have sought a private diagnoses is because they disagree with the NHS one. Some of them may have been right to disagree, but I know for certain some of them weren't.
Think, OP, before doing it. Think who might you reasonably guess who might donate. Your parents? Siblings? Friends? How many might donate? I think I might get 10 people. That's £200 each. I don't think I could expect any except my parents to donate anything close to that. I think I'd be lucky to get £500 in total, probably closer to £200-just giving takes some of that as commission too. Is that worth it?
So what would make me donate:
You need to say exactly where it's going-I don't want to donate to "support medically" and find it actually went on "a new bathroom because he didn't like the colour" (genuine case I read once, child in question was less than 18 months old).
You also need to say why it is needed over the NHS. DD is missing her hand. I've seen a number of appeals for prosthetics, often exactly what the NHS offers, and, what's more, for upper limb prosthetics very few children choose to wear them beyond the first couple of days once the novelty has worn off. I suspect a lot of the time they get their first prosthetic funded privately, do more publicity, then quietly sidle off with the spare money and use the NHS/don't get another. That could be me being cynical.
I'd also want to know where the money will go if you decide not to go ahead with it (or in the unlikely event of getting too much). Best thing is to refund or a charity. If you can't find a relevant one you like, try a local one that people will know about.
Don't hassle people. Yes, make people aware. But no shaming/naming on social media. And don't post it three times a week.
And don't go for too much of a sob story. I'd rather it was factual. That may be a personal dislike, but a sob story doesn't have to go too far in for it to start sounding like you're playing the crowds, and becomes unbelievable.
But even with above, I'd only donate to a close friend. I think there's probably about 4 people other than family where I would even stop and look further than scrolling past. They're all people that wouldn't normally ask for money, would have investigated thoroughly, and would give you the moon if you asked for it.