All my tubs and sinks in the bathrooms have an 'overflow drain' near the top which you are unable to fill beyond. Is there any reason why that wouldn't be sufficient?

In Japan it's very common for bathrooms to be wet rooms; likewise South Korea and Taiwan among others.

But USA bath tubs have overflows (redirecting water into the drain if the taps are left on), surely?

The tub sits in a big shower booth. Other bathroom amenities like sink and washing machines are installed outside the booth. Toilet is usually a separate room entirely. So no problem overflowing water onto the floor since that's how you take a shower.

Bathroom floor drain is common (might even be a code requirement?) in commercial buildings but not in homes.