What tax benefits do people gain from donating to 501(c)(3)s?

Isn't the tax system progressive, so that your first (rough numbers from memory most likely inaccurate) say, $12000 gets taxed at X%, while your next, say, $8000 gets taxed at Y% (Y>X), and your next $50000 gets taxed at Z% (Z>Y).

The term for this is "marginal tax rates".

In this case, supposing you had earned $22,000 in a year and then donated $2000 to a charity in the hopes of saving on taxes and taking home more, it doesn't actually have any effect on the rate at which your remaining $20,000 is taxed at, whether you donate or retain the additional $2000, right?

Wrong.

That is because all deductions (including charitable ones) reduce your Adjusted Gross Income, which is calculated next year during tax season after you've earned all your money.

IOW, no matter when in the year you contribute that $2000, for tax purposes it's taken "off the top".

(However, the person contributing $2000 to charity won't get any tax benefits because you only get a benefit if you itemize deductions on your tax return instead of using the Standard Deduction. But the Standard Deduction is $12,400 for Singles.)

when rich people make contributions to tax-deductible charities to benefit themselves in the tax code, how does it actually benefit them?

See above.

There is no net gain. I am in the 22% bracket and when I donate $1000 to charity, I see my net tax bill drop $220.

When I ‘donate’ to someone in need directly, there’s no such deduction. Which is why it’s better from a tax perspective to use a registered charity.

Similar to how I look at mortgage interest - the tax deduction can be a discount on the mortgage rate but not a reason to get a mortgage - donations should be made because you are so inclined or because your religion commands you to do so, but not “for the tax deduction”

You basically have the the concept correct. If you donate $X to a charity it will save you Y% of your donation in taxes. Donating money, or stuff, to a charity never saves as much as the value of the item.

There could be a few very rare edge cases where donating money or stuff gets you under a threshold, but since the the tax deduction takes place late in the tax calculation process it doesn't apply to very many people.

I view the ability to deduct a charitable contribution as a bonus. It makes it easier to afford the contribution. When the tax changes were made in late 2017, it moved me from somebody who itemizes to somebody who takes the standard deduction. I could have upped my charitable deduction significantly to get back to being a itemizer, but it would have taken thousands of dollars to get back over the threshold, and would have returned very little monetary benefit.

I predicted that donations to some charities would drop after the tax changes. They did. It will be interesting to see what happens to charities in 2020/2021 because of the crisis.