Or set 12:00 to noon. People colloquially call 12:00 'noon' anyway, even though most of the year 1:00pm is noon. (In fact, this is so common that I'd probably better define noon: the time during the day when the sun is most directly overhead, or the time when the sun is either directly overhead or due north or south of that).
Yes, there are complications, which is why people set standards. Successive noons can actually occur slightly less or slightly more than 24 hours apart, so you standardise on the average noon. (This is where the ‘Mean’ comes from in ‘Greenwhich Mean Time’.) And even then, you have to change your clock every time you change longitude just a little west or east, so instead of using local mean time, you use mean time for a nearby large city centre or other landmark. With long-distance travel, it becomes convenient to divide the earth up into Time Zones, each one hour apart (although some places like Newfoundland like to mess with this).
Already with Time Zones, governments began shifting the boundaries westwards (sometimes severely) in an attempt to get people to wake up earlier. (Newfoundland is a perfect example; you'd think that it has an odd Time Zone because it's straddling two of the standard ones, wouldn't you? But it's not! It's entirely within what should be the Atlantic Time Zone.) Then they add Daylight-Saving Time on top of that. They are no longer setting standards; they are trying to control behaviour.
In the long run, of course, it doesn't work. People in Nebraska (where I grew up) are lobbying to have the school day start (and end) later so students can get a better night's sleep beforehand. No surprise, since the Time Zone boundaries are shifted about half an hour west there and half of the school year is in DST to boot! The call to start school one hour later actually matches the average effect (half an hour early in the winter from being on Central Standard Time, 1:30 early in late spring and early fall from being on Central Daylight Time).