This falls under my rule #1: Dont ask for a rule, you’ll eventually regret it.
I would encourage the SM to handle this on a scout-by-scout basis. And leverage the situation to encourage personal growth.
Is your troop meeting every week in the summer? If so, then maybe issuing a couple cards a week based on the how well the scout is acting on contacting counselors and working on badges is a good idea. That way, the scout can explain why he’s stalled with some badges and wants to try to start others.
If the troop is not going to be that active, or the scout will be traveling, then maybe the scout should get as many cards as he plans to start in the summer. In the fall, follow-up with scout to see if counselors for each badge were contacted. If they weren’t, then have a conference with the scout about the first and ninth points of the scout law.
Regarding the Advancement Chair … of the 13 responsibilities listed in Guide to Advancement Section 188.8.131.52 “try and help the scout complete what they have …” is not one of them. Now I know that GTA says that list is not all-inclusive, but it’s tone is clearly adult-facing, not scout-facing. The committee works best when they replay with “yes, but safely.” And, so far, I don’t think there have any incident reports of scouts harming themselves from amassing a stack of uninitiated or partial blue cards. So, from a committee perspective, I’d let the SM and ASM know that you’ll support whatever they think is best for each scout. I would especially encourage them to treat each scout on a case-by-case basis. Then, in a couple of months, ask that scout who got the 21 blue cards that you’re ready to log his progress! Also, I think it’s fair game, if you wind up sitting on his board of review, to ask him where he is with all of his blue cards.
Bottom line: instead of making a rule, have some fun interacting with your scouts!