Do induction cooking tops work well on batteries (or generators)? IIRC our induction plate has two-phase power because it can pull more than 3.6kW.

Sure, as long as you size the system to expected loads.

An 8kW generator suitable for occasional use is only ~1,000$. A Powerwall 3 does 11kW continuous and peaks at 30kW for transitory loads like starting heavy equipment.

The most convenient solution where a generator automatically kicks in during a power outage requires an electrician and extra equipment, but there’s also real tradeoffs to having gas lines going to your home.

There are models that include a battery to reduce the input power requirement. That's not quite the same as the question, but it answers it, you just need a big enough battery and they are fine.