vecollect is the letter collection service my school uses. It cost 20 dollars to use and basically you plug in the info of your letter writers and make 'quivers' which you'll have one for allopathic and osteopathic and then you'll add the letters to the quivers and then select the schools you want to send it to. It's not too bad of a service, but it seems sort of beta to me and is a little tricky to use. You can also use it for other professional schools such as dental, podiatry etc.
And OP that seems like a good plan with addressing the letters and then just having them send them off. Perhaps have him write his signature over the flap of the envelope so that is clear that even though the address is in your handwriting, you did not look at the contents.