A friend just happened to randomly ask me essentially this same question this morning after coming back from an early morning run.
[For simplification of course, I am ignoring all cases that involve the effects of wind/weather fronts, rain, cloud cover etc. and focusing on your "ideal" no wind, no cloud cover situation]
The question itself can be a little misleading depends how it is phrased however. Compare the subtle differences in how people can essentially be asking the same question:
"Why does it get colder when the sun begins to rise?" kind of sounds like the sun rising itself is making things/causing things to become colder and this is what needs to be explained, which is not the case.
Whereas rephrasing the question as "Why does it
still get colder when the sun begins to rise?" is a much better (and accurate) way of phrasing the question and indeed the actual observation.
Now the other thread titled
"Temperature begins to fall at dawn" can itself also be considered a poorly phrased "question" with the OP asking "I heard somewhere that temperature actually begins to fall at dawn and then goes up later on. Why?" The use of the word "begins" sounds like they are saying that up until sunrise, the temperature was steady, but when sunrise hit, the temperature began to drop and they want to know why. Well that may indeed have been an actual observation but I doubt it was one that was devoid of wind/weather front effects (convective), possibly also combined with cloud cover effects. However, from reading the OPs later posts I am sure this observation was not what they were asking about.
Semantics aside, I think we can agree that everyone has been asking the same question/trying to understand the same phenomena (one that does not involve any wind/weather front or cloud cover effects) and I believe adequate explanations have been given.
I will however propose an alternate way to explain/look at this situation. Radiant heat transfer is difficult to "see" so I attempt to construct an analogy to perhaps provide a better "visualization" for what is going on.
Consider the universe like a very powerful vacuum cleaner sucking heat from the earth. Consider the sun like a very powerful hairdryer blowing heat towards the Earth (Yes, ignore the fact that convective heat transfer is NOT primarily responsible for how the sun heats the Earth and how the universe cools the earth). At night, the sun is obscured and it is just the universe continuously "sucking" heat away from earth, making the temperature continually drop.
There are other "suns" in the sky at night however, we call them stars. They too are like our sun, a hairdryer blowing heat towards us but they are so far away their power is nothing compared to the power of the universe sucking heat away from the earth, or to the power of the sun! Now consider what happens when the sun begins to rise. It's like gradually putting more stars in the sky (or more hairdryers pointing towards the earth, or starting to reveal a very powerful heat blowing hairdryer). Would you think that it makes sense to assume that as soon as you catch a glimpse of the sun that the temperature on Earth should immediately drop? Of course not. When you think of it like I have explained, it seems silly to expect that instantaneously the Earth should start getting warmer. You need to reveal enough of the sun so that the power (rate) of heat being blown towards the Earth just is enough to counteract the universe's opposite (rate) effect of trying to suck heat away from the earth. For earth, this apparently happens shortly after sunrise. This is essentially the turning point at which the rate of heat transferred to the Earth exceeds the rate at which it is being sucked away by the universe.
Conversely, understanding why temperatures are hottest not at peak sun but a few hours after can also be used by this analogy and the subsequent net direction of heat transfer at any given point in time.
Curiously, a better exploration for what is going on here can be had by considering "what if" scenarios. For example, what if the sun "stopped moving" in the sky when the temperature in the morning just started to begin to increase? Would the temperature also stop increasing?
This question itself makes it apparent that an understanding of "steady state" thermal systems (the concept of thermal equilibrium and time constants) is not only important to answer this new question, but to also answer the original question in more detail, which is in fact a much more complex/dynamic thermal situation to fully explain.
Answering seemingly more simple "what if" scenarios can go a long way to building an understanding on how to answer or approach answering a more complex question.
eg. How would temperatures on Earth vary if the Earth actually rotated faster/slower than it does?
This great video is a perfect example: