My book wasn’t trying to build a rectifier.
It says this:
It is evident that AC current does not really flow through a capacitor. It flows into the cap on the first alternation and charges the capacitor plates. Stored, it is returned to the circuit on the second alternation as the capacitor discharges.
So AC makes a lamp glow. AC circuit with a capacitor makes it brighter.
According to this teaching illustration. But maybe there’s a reason that isn’t what we see in the practical application of AC to light bulbs on mopeds?