[GregB]

..... Since I have an entrance box on the bottom with a clean out door, and the third box that was recently added has minimal construction on it so far, I put the broken comb on a waxed bit of cardboard and slid it into the entrance box, figuring the bees would salvage what they could and I would remove what was left in a couple of weeks.

Was this a good idea? or should I just pull it out and let the bees do as they like?

thanks for your advice

Since you do not show your location (and the appropriate season along with it) no one will be able to give you a good answer.

With that, it is better to keep that broken comb inside for the bees to cleanup, not outside (to prevent potential robbing trigger).
As a matter of fact, I would just attach the comb back to a top bar using painter's tape and just put it all right back into the hive (could be broken up into few pieces too and attached to few top bars).
You can also put those re-attached combs into your "third box that was recently added has minimal construction on it" - this will make the third box more attractive to the bees.

OR have some honey for yourself - the no-brainer option (I often take advantage of collapsed/malformed combs to just steal them for myself).