You could, but you'd have to keep the money in savings anyway, and there'd be some hassle at a time when you might not want to deal with hassle. In practice, it turns out most people just pay their premiums, at least in Japan.

But if it turns out to be a problem, you could always charge interest.

I think a baseline requirement for these types of systems is that people don't go out of their way to be assholes for negligable gains.

My understanding of Japanese culture is that, by and large, people don't do that. Americans, meanwhile, have a pretty heaping helping of petty asshats. For exhibit A, go look up some of the asinine BS tax protesters try to get out of pretty basic civil responsibilities.

I wonder what they do if someone can't pay.

The system doesn't seem too different from the US to be honest, in that people with poor financial planning skills may decide to skip on insurance premiums and end up not having enough money saved to pay for medical expenses/back premiums when they occur.

Be that as it may, it sounds like the Japanese system is much more effective at keeping prices low.

I'd guess there's some kind of interest/penalty you have to pay along with the back premiums.

Then they’ll add up the premiums?

Unless there is interest or a penalty then you can take all the premiums you've saved + interest and just pay them.