There's no need for the mains frequency to be a multiple or fraction of 60. Before standardization, anything from 16.75 to 133.33 Hz was possible.
There are probably historical reasons why we ended up with 50 Hz or 60 Hz, and why Japan ended up with both.

Split phase transformers, stinger leg transformers? All sounds complicated to me.
You could argue that we have a simpler system in that we all have three phase power coming into our homes (even my apartment that was built late 1950s has it).
Only three phase transformers in the grid.

Typically in homes only one main fuse was (and still is) installed to get single phase 230 V between L and N. Most houses are still like that.
But more and more connections are being converted to threephase power for power hungry devices like heatpumps, jaccuzis, spas, jetstream pools, photovoltaics etc.
For about 300-350 euros the utility company fits two extra fuses and changes the kWh meter for a threephase model. Then you can have an electrician change your distribution board to add threehase circuits.
Existing single phase circuits can be run from one of the phases, or spread across the phases to get a more equal loading.

New houses will have threephase power installed by default, even if not used as a 400 V threephase source.

For homes that's it. Industrial properties with heavy machinery can also get 400 V/690 V connections. Even heavier industry will probably be fed from 11 kV and install their own transformers on the property.