There's no reason you have to kill the 12. It would work to just have 1:00 be the first hour of the day, have the day be 1:00-12:59am followed by the same as pm. 1:00pm is "meridiem". No change to analog watches.
Obviously there's zero chance of switching to that at this point from a pragmatic point of view but it does just work out fine.
Given an AM/PM 12 hour system, unless it's obvious from context, the best thing is usually to write noon or midnight. 12am and 12pm in isolation can absolutely be confused. Even if I know the convention, I don't know if the person writing 12am does.
Obviously there's zero chance of switching to that at this point from a pragmatic point of view but it does just work out fine.