When I sign off students for their first solo, I usually jump out of the airplane and watch them do their first 3 landings. It's more of a motivational thing, than anything else. I want to see the look on their faces when they successfully complete their first solo. It's rewarding as a CFI.
When I was a new flight instructor, I only allowed my students to solo IF I was at the airport. I wanted to check the METARs, TAFs, radar and look outside, just to be sure that it was safe for them to go. It's not that I didn't trust my students, I didn't trust their experience level.
Now after 2000 hours as a CFI, after the first solo, I only ask my students to text me before they go up, and text me after they land. This is not required by any regulations, but it's just something that I do... to prevent some random phone call from my Chief Flight Instructor or the flight school owner, or the tower. IN CASE anything happens, at least I know who was flying, since I have multiple pre-private students with solo endorsements.
To answer your question - No, your instructor is not required to be at the airport as per regulations, however, IF your Flight Instructor only wants you to solo while he/she is at the airport, then it's his/her prerogative. I understand that this may not be in the regulations, but when it's the CFIs signature on the line, he/she gets to make the rules regarding crosswind components, ceilings, etc. Your instructor may not be comfortable letting you solo unless he is at the airport to check the weather conditions himself. Since it is his ticket on the line when you solo, as a student, you should respect what he is or is not comfortable with. If you can't, then your option is to find another CFI to sign you off to solo.