Did Spock attend Starfleet Academy?

I have been reviewing what we learned about Spock and
Sarek from the first half of the Star Trek Discovery season, and now I am wondering whether Spock actually attended Starfleet Academy.

I have always assumed that he did but it has never been explicitly stated (except in the JJ Abrams movies, which are a different timeline).

Here's what leads me to this conclusion:

We know that Michael Burnham graduated from the Vulcan Academy of Science circa 2249 or 2250.

The episode Lethe tells us that Sarek secretly chose to keep her from joining the Vulcan Expeditionary Group (i.e. the Vulcan space service) in a deal to allow Spock to join instead.  The Vulcan snobs didn't want two human/half-human people in their space service and gave Sarek the choice of one or the other.  Spock, of course, did not take advantage of this opportunity and joined Starfleet instead. 

Sarek turned around and found Michael a post on the USS Shenzhou.  Within two years, Spock was a lieutenant on Enterprise during the Talos IV incident.  He prosumably could not have attended the academy AND risen to the rank of lieutenant in that time.

That implies that neither of them attended Starfleet Academy, because they got the equivalent of their college education on Vulcan, and that they are about the same age.  He may even be a couple of years younger than her.

Note that Sarek is revealed in Discovery to have sought a position for Spock in the Vulcan Expeditionary Group, however in the Original Series episode Journey to Babel we were told that Sarek was upset with Spock for 18 years because Sarek wanted Spock to join the Vulcan Science Academy, which is presumably something different.

I had always assumed that a) Spock was quite a bit older than he appeared, several years older than Kirk, and that he did attend Starfleet Academy.

Now it appears that he did not, which resolves questions about Spock's statements in The Wrath of Khan that he never took the Kobyashi Maru test.

One of the things I have always enjoyed about Star Trek has been the way the over 700 TV series episodes and the movies actually hung together in their continuity. 

I wonder if the Discovery writers KNEW what they were establishing on Star Trek continuity with respect to Spock and the Academy?