In theory it does work, and it has historically worked in the past. A lot of wealthy people in the last few centuries were philanthropists that built schools, hospitals and other public works.

The main deterrent these days is that your typical billionaire is greedy and entitled.

Plus as we saw with some merchants and colonial figures, your name could be scrubbed from the history books and statues of you torn down if your past actions are incompatible with modern day morals. Edward Colston is a good example because despite him pumping a lot of money into philanthropic projects, he made his fortune from the transatlantic slave trade.

I can almost guarantee that people would have much more favourable views of Jeff Bezos, Elon Musk, Bill Gates, etc if they pumped tonnes of money into building new homes and actual public transport infrastructure.

Starting a private space company doesn't count as philanthropy. As for Bill Gates, years of medical disinformation have built up this narrative that he's pumping money into medical research and vaccination programmes for nefarious reasons, like planting microchips into people.