No because the year last for a longer period, while you still need to know the immediate context for what day it is..
A year is really not a long time.
The most important context is the year.
How old are you?
How old is your car?
How long have you been married?
When did grandma die?
When did you move into your house?
How long did World War 2 last?
When did this economic cluster suckitude we are in now start?
etc.
For almost all important dates and time measurements, the year is the most important time measure.
Really? You mean, when people talk every day, they ask, "What year is it?" rather than "What's today's date?"