Intuition depends on experience and context.
My guess is that what is intuitive for time keeping has changed over time (no pun intended). Consider back when most people were living directly or nearly directly off the land in single families or small groups, working very hard to grow enough food or catch enough fish or game to keep alive.
I think that they would use three references for time: sunrise, noon, and sunset. Sunrise would be the main one, because that marks the time when the dangers of the night go away, and your domestic animals wake up, and you have light to start working your fields or hunting your game. I think noon would mostly simply be used for pacing--it marks when you have used up half of your daylight, and so is a good time to judge if you are working at a good pace or need to pick it up for the rest of the day.
For such a person, if they had a use for a clock at all, a sunrise synced clock would be intuitive because it would put the things they do at relatively fixed clock times each day. For example, if it takes you about four hours working the fields each morning to get hungry enough to need a lunch break, then on a sunrise clock that has you starting in the fields at 8 AM everyday, you are coming back to the house for lunch at 12 PM every day.
On a noon-based clock, you would be starting in the fields at a different time each day, and so if you came in after four hours you would be arriving back at the house expecting lunch at a different time each day.
For people at that "barely making it off the land" stage, pretty much everything they might need timekeeping for is tied to sunrise or sunset in a similar way so those are the intuitive references for time.
It's not until you get a large enough and diverse enough society that you have people doing significant things that are not directly tied to just staying alive off of the land, and doing things that involve synchronizing with people who are traveling a long way, that noon-based starts to make a lot of sense.
As settlements become bigger and bigger and more specialized, the dark becomes less dangerous, more work is done indoors by candle light or lantern light, and syncing to the sun becomes less and less important. More and more of our time is dictated by the clock and noon-based makes even more sense because it is clock friendly.
But now we've reached a point where we have enough leisure time that we actually do want some sun synchronization. We want to go to work earlier in the summer, when the sun rises earlier, so that we'll more time for play in the sunlight after work.
The most obvious way to do that would be to change work hours. Normal offices hours would change to 8 to 4 for the summer from the normal 9 to 5, and we would not fiddle with the clocks.
I suppose this was not done because it require too many people to act. Instead we diddle the clocks, so the clock says you are starting work at 9 in the summer, but it is astronomically an hour earlier than the time you started work the rest of the year.
What I'm suggesting in my earlier comment is that now that clocks are actually computers that can do quite sophisticated calculation, we could use a psuedo-sunrise based time has both the daylight maximizing property of daylight savings time and has the reproducibility and consistency of noon-based.
You can think of it as a kind of continuous or near-continuous daylight savings time, where instead of adjusting the clock with a big jump twice a year, it is adjusted either continuously or in a small step every day.