Is it sexism to charge higher insurance rates for new male drivers with no record?

Yes, they are, at least in states that allow it. In many states elderly drivers pay more for car insurance than younger (but older than age 25) drivers do.

I believe the clearest example of this is what is commonly called being self-insured. You pay nothing and accept all of the financial risks, be it collateral (property) or cash. Or you have very high deductibles and lower your premiums by transferring only the risk of a catastrophe – some event that is unlikely but would be devastating financially if it occured. There is no purpose in buying insurance if your aren’t transferring the volatility of the risk. Without that transfer of risk, those agreements are just loans.

In California, at least, you are allowed to post a bond rather than carry liability insurance.

Presumably, if you didn’t have enough liquid assets, you could take out a loan to post that bond, but the economics probably don’t work out very well.

That’s factually incorrect.