Are options a form of insurance?
No, the primary purpose of options is not to provide insurance against changes in the price of the underlying instrument: options don't have a primary purpose, they don't have an agenda, and they don't have a plan. They're just another tradeable instrument.

Some people buy them as means of insuring a position. Some people write them as a means of insuring a position.

And as Aurigae notes, in a crisis all correlations tend to one; so even if, in theory, your options insure the rest of your position, in reality, in a crisis, the counterparty risk can become very high, so the insurance is least available when it's most needed.