what's the helmet situation in other countries, especially around ad-hoc rental scooters? I see scooters and bikes for rent here, but no buckets to wear, and I wonder the legality of that too.

Outside of Australia, helmets are not mandatory and data from there shows it has had a negative effect on not only the numbers of cyclists, reducing "safety in numbers" but also an increase in children with head injuries, because drivers are less risk adverse ("Mary Poppins effect" - drivers give more space & make less dangerous passes around cyclists in everyday clothing).

The most simple solution to increasing cyclist (and pedestrian) safety is to build better infrastructure. Design roads so that vehicles have to slow for junctions and roundabouts, especially where there's mixed modal traffic. Prioritise active and public transport over inactive modes. Make cycle, walking & bus routes more direct and make driving routes more circuitous. Discourage driving into town centres and stop building out of town retail parks.

Back to the topic in hand:

If anyone wants to claim that "helmets save lives" then why aren't they advocating them for vehicle passengers, who experience greater numbers and severity of head injuries than cyclists, or for pedestrians who have about the same number, possibly slightly more than cyclists?