As far as I know, all Avelus starts with the burial, not with the actual death.
Particularly, why saying Kaddish does not start with the moment of death? And is it just a general Minhag or there are some secret matters?
Well, since Kadish is said by the mourners, and they are exempt from davening, they won't be in shul to say Kadish, and may not even be allowed to say it (same way they cannot daven or bentsch.)
That said, at a funeral the Kadish is already said, before the burial.
Also, if one has given up hope of finding somebody presumed dead (e.g. they drowned and were never located) then one can usually say Kadish for them, even though there was no burial. See Kitzur Shulchan Aruch 204:4 for details.
Also ibid 204:3 we see that one can start all Shiva procedures before the burial, when the burial is out of town.
We also see from the Kitzur 196:9 that Kaddish can be said on Shabbat if the burial is delayed until after Shabbat.
So, it seems - like I already postulated - that the mourners usually don't say Kaddish before the burial because they are not allowed to be doing any positive Mitzvos; their full-time Mitzva is to deal with the burial.
Who exactly is qualified to say so and so has just experienced "the moment of death"?
Nobody, in good conscience, can testify to something he or she is unsure of; therefore, the waiting period (even if less than 24 hours) before the burial is a way to be certain that one is not wrong on the issue.
Isn't that simply a practicality, that Kaddish is not mentioned at the moment of death but at/after/with burial?
Therefore it has become "easier" to diagnose somebody as dead at/after/with burial rather than at the moment of perceived death?
In secular first world countries, like Great Britain, for example, "there is no legal definition of death" (www.bbc.com/news/health-19968625). There are only "guidelines for the diagnosis", which therefore opens the door for a misdiagnosis, which is ripe in second and third world countries.
Doesn't even the Torah (for example Devarim 31:16) uses the verb to sleep/lie-down when somebody dies because by sight, before burial, death and sleep look the same?
Devarim 31:16 And the LORD said unto Moses: 'Behold, thou art about to sleep with thy fathers; and this people will rise up, and go astray after the foreign gods of the land, whither they go to be among them, and will forsake Me, and break My covenant which I have made with them.