Two issues:
You have to bring your own, which is a relatively large thing you have to remember to bring and store. Either people forget them, or can't be bothered... because the whole point of the scooters is ad hoc transportation if needed when it isn't expected.
You get one provided (on the larger ones that include them), but now you have a singular helmet which may have just been used by a sweaty gross human. And it just got put away into a sealed box.
The only thing I can think of to alleviate both of those fears (and still allow Scooter sized things to exist..) would be some sort of helmet vending machine with a 'used helmet' return box. It'd require way more helmets and some sort of cleaning mechanism, but at least that would solve or alleviate the two main concerns. It does immediately mean you now need scooters in close proximity to vending machines though...
For people to actually carry one, it would probably need to pack up to the size of a tote umbrella.
To be honest, I think helmets are good, but a mandate has its own issues. I think usually the culprit is extremely bad infrastructure for not-cars; few places have bike lanes which are safe and comfortable to use and in good condition. Or bike racks which might be a logical place to otherwise put scooters. Sidewalks are usually not a good substitute, and even where they exist often they may not be ADA compliant in terms of ramps or width, they are bumpy due to tree roots etc. displacing the sidewalk panels from their original position and creating unsafe bumps, etc.