On a car (at least an old one) the ridiculously robust lead-acid battery is used also to filter out spikes and to act as a voltage regulator. If you've ever wondered why the headlamps on a kick-start moped flicker while idling, that's what you get without using batteries as a filter. Li-ion wouldn't survive very long in this application AFAIK.
There are plenty of replacement LiFePO4 auto batteries available, and they work very well without needing much in the way of support circuitry. I think that it's mostly cost keeping lead-acid around (as well as operating temperature range and lifetime), not that nothing is suitable for the application.
The headlamps on a moped probably flicker because it uses a magneto instead of an alternator as its power source.
Can't comment about mopeds, but in car, when your battery nears the end of its life, when you turn the steering wheel (with electric power steering) while stationary, headlights dim for a short moment until the alternator increases its output.
