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I would say it depends on (for both construction and farming) the type.
With farming, that sounds like something similar to cattle (meat) or crop farming. I grew up working in construction (electric) and worked a few times throwing hay (worst job ever). With farming, throwing hay never needed to be done at four in the morning, in fact it seemed like the A-holes went out of their way to ensure it was done during the hottest part of the day. My buddies (grew up on their farms) and had a couple hours of work to finish before they came to school. I used to feel sorry for them until I realized that by 22 they were making 100k.
As far as construction, I have worked nights as well but that was typically the parts that were indoors (mainly factories), I seem to remember (I may be wrong) the electrical companies that had night shifts outside were mostly very large outfits. We typically reserved outside work to be done during daylight whenever possible.
While it doesn’t always work out this way (geography, time of the year) we call it midnight because its supposed to be halfway through the night. If so then after midnight we would expect the sun is now coming back towards us (relatively speaking) so it should be the beginning of a new day. >

Right, but why are we saying good morning when we still have half of the night remaining? >

I don’t think people typically do this automatically. Rather we begin doing so after we wake up regardless of if that is at 3am or 6 am as our new day has begun for us. >

Maybe it's different other places, but I've heard plenty of people say things like "I was out until 2 in the morning." >

No, you are correct, that is common. I guess we are discussing two separate things: when the new day starts, and when it is appropriate to start using the term morning versus night. An alternative is when people say that they stayed up all night, I don’t think they mean until midnight. >

Agreed. A more important aspect of my argument is work hours. Instead of a 9 to 5, you'd have a 3 to 11. But you say "my country already solved this with a 24hr clock."  Well, night shift workers would typically come in to work and leave on the same calendar day.  Very few jobs have hours that lap both sides of 6am. Tons of them lap midnight. >

I understand but would also suggest that the exception here is also the historical precedent for the way we measure the work day (although it has substantially changed with electricity) and that is farmers. I grew up in a farming town, and for most of the year they were already working substantially earlier than 6am.
The reason I bring up farmers is just to suggest that those jobs that can be arbitrarily arranged such that they fall in the same day are typically jobs that do not rely on forces of nature as much. Jobs that require natural daylight (farming, construction, et cetera) have historically been more important to us than jobs where artificial light is acceptable. Just my opinion though. >

I'm in construction. Sometimes we come in before 6, but rarely. Most jobs start somewhere between 6-7. If we run a job around the clock with night and day shifts, those shifts are typically 6a to 6p. 
Is farming typically an hourly job? I did more ranch related things growing up, but hay paid by the bale, and tractor work was by the acre.