There's a clear logic to moving from 11.00 am to 11.59 a.m. to
12.00p.m.

1.  The instant noon arrives it becomes afternoon.
2.  p.m. (post meridiem) denotes afternoon, which it has become even
before 12.00.01 arrives.
3.  At the other end of the day, the same considerations apply: as soon
as midnight arrives, it becomes a.m., long before 12.01 a.m. arrives.

As for having options: it was the arrival of steam locomotion and
railways which led to the standardisation of time. Can't really see it
falling apart again...

Alf