That's a very naive understanding of how charities work. You don't donate to a charity, you donate to your charity.
Now, obviously your charity still needs to act as a charity so that money can't go back in your pocket but there are plenty of things the charity might spend it on that are in your financial interest and because you founded it, you likely have sufficient influence over it to make that happen even if you don't formally personally make its decisions.
Also, if you donate $100 in stock to a charity, that deducts your taxable income by $100 but it doesn't cost you $100 in income. Arguably a more egregious example for this is high art where you can create and destroy value through auctions (i.e. the value of your donation may be massively inflated compared to what you paid for it).
No, he was definitely guilty. But if you do good things in recompense, it's easier for people to let it slide. As with modern antitrust, labor rights infringements, etc., it's not so much, "Are they doing it?", as, "Do we care enough to investigate and build a case?". Bankrolling a public good that politicians know need to happen but don't want to go through the political pain of appropriating funds for makes them care less.
Given that the US isn't a perfect capitalist system (i.e. what supports of capitalism often refer to as corporatocracy or crony capitalism) why would you assume that he didn't do the closest thing to illegal that he could get away with if it was most profitable to do so?
That said, the actual reason for his philanthropy seems to be fairly well documented: he did almost all of his philanthropy in the last two decades of his life (i.e. past what we would now call the age of retirement) and he was regarded as a shrewd business man up to that point and is now instead remembered as a generous philanthropist instead. So in other words, it was a very successful PR campaign to establish his personal legacy to survive his physical existence (i.e. "to secure his legacy" as the comment you originally replied to stated).
That said, he was also supportive of the ideas of progressive taxation and an estate tax, so the philanthropy may have simply been an act of last resort to part with his money when the government won't take it away through taxation. In that light it sounds more like his philanthropy aimed to meet the needs the welfare system failed to address. This is very different from the kind of philanthropy most billionaires (or "tycoons") get/got up to who are often very much against being taxed more (sometimes even despite publicly stating the opposite).
Do you have a cite for that?
BTW, donating to charity is tax deductible. For example, if I donate $100 to charity, I can deduct $100 from my taxable income.
My choices:
1. paying taxes: I pay $20 2. donating to charity and deducting it from my income: I pay $100
I'm $80 worse off financially by donating to charity rather than paying taxes. As a tax avoidance scheme, donating to charity doesn't deliver.