Howjado - I just wanted to say I understand your feelings and don't think you are being silly at all. I would be the same.
I have been very lucky and been present at the birth of my grandchildren, along with the father. My husband (the grandfather) did wait in the car park, but he came in within minutes after the births too. Knowing how wonderful a experience it was to share, I would feel a little upset not to experience it again someday, but of course understand it is for the couple themselves to decide.
The other grandparents visited my DD/SIL the next day, but it was a week or more before other friends or family visited as it was then they needed their time together as a new little family.
I hope all goes well for you. If you are like me you will fall head over heels in love, and be needed often in the weeks and months ahead.
I've got another 'keen'... Ouch!