Why is the time of a person's death treated with such importance?

to keep the burro crats happy.

For a number of legal reasons, but I think mostly because it determines precisely the creation and extinguishment of rights and liabilities. In cases of succession for example, that determines the exact (up to the minute) time the property and obligations of the deceased are transmitted to his heirs by operation of law. The exact minute thing seems so unimportant at first, but details like this are magnified by law when people's lives, liberty and property are at stake.

Many husbands and wives have clauses in their wills that state the first one to die leaves everything to the surviving spouse. However, the second one to die leaves everything to the that person's kids. With so many people being divorced and remarried, the husband's kids may not be the wife's kids.

Now, many husbands and wives travel together and die in the same accident. Courts were swamped with will challenges trying to determine who actually died first. (Dad's kids would sue Mom's kids to get the money.) To free court time rather than hiring more judges, the legislatures adopted rules stating the time of death must be noted by the attending physician. Most of the will challenges ended at that point.

Later it was learned the time of death was also useful for investigators to determine who was where in cases of murders.

Vital statistics offices have to keep a record of all of this stuff, even the time of your birth. Also, if the death was suspicious or a homicide, then the time of death can be critical in an investigation.

Forensic medicine, legal veracity, and research in various medical fields are the most glaring reasons to determine, as accurately as possible, the time of a persons's death. Check further from my source. Good Luck !

Just like your birth time is of importance.

why was the time of your birth of such importance?

Its just record keeping.....time of birth is also carefully recorded

because the american government owns your body, life, and soul.