So imagine there exists a charitable billionaire that wants to do good. How in your eyes would a billionaire go about donating their money without drawing this same criticism?

I'd say the answer to that, is that they should simply give significantly more than what they're currently giving. We're talking of people who could easily give away 99% of their wealth with 0 personal sacrifices. If they're giving less than 0.1%¹ instead, I just want to know why?

I found this webpage extremely helpful for putting into perspective just how much good they could be doing: https://mkorostoff.github.io/1-pixel-wealth/

¹) That number probably needs to be a lot smaller, but I don't want to make any claims.