What is the advantage of partition Date field into individual Year(int), Month(int), Day(int), Time(time) columns?
I guess it depends on how you want to use that data. There may be some advantage to the application using it. For example it might make reporting or filtering by year or month easier.

But I've never heard of that as some kind of best practice on its own.

The downside of this approach is that you lose the ability to use the standard date functions in SQL. Simple things like duration between 2 dates or times is simple using datetime fields but a nightmare with ints. You also lose the inherent validation of a proper date datatype. Ask your  boss if there is anything stopping you from inserting 30th of February or a 13th month.