If I recall correctly, in the old 'FS' days if your destination was an aerodrome with a Tower, your SARTIME was held by the Tower.... AMMBZT in FS/ATC speak of those days before the Y codes came in! At some locations depending on local procedures it was in fact held by the briefing office. At remote locations the SARTIME was held by the FS unit responsible for that area. At some locations ATC did not believe it was part of their job and would strive to have FS hold the details. But yes, times have changed.
At the end of the day how the SARTIME is managed should be transparent to the pilot. He just needs to submit it, and cancel at the appropriate time, and for search action to be commenced if he does not.