Because it would be too easy to game the system.
For insurance to work to mitigate risk, you have to pay a rate that is roughly the expected cost of insuring you (i.e. the probability of illness * the cost of that illness). That requires that you commit in advance to paying before you know for sure whether or not you are actually going to get sick.
If you could avoid paying your premiums for the first 10 days of the month and then only pay 2/3 of the premium for the remaining 20 days of the month, there would be a large incentive for people to game the system. If they were healthy at the beginning of the month, they'd skip the payment. If they started feeling poorly after a week, they'd pay a lower premium and immediately go see a doctor. If they felt fine all month, they'd skip the premium entirely. The insurance company would quickly go out of business if a large fraction of their customers only paid when they were going to immediately seek treatment. They depend on all the healthy folks paying their premiums to cover the unlucky folks that get sick.
Obviously, not every illness is subject to this sort of gaming. If you're hit by a car, it's not likely that you're going to be able to delay treatment until you've paid your premium. But enough illnesses are subject to gaming that insurance companies have to prevent people from doing so in order to remain in business.