Maryly,
You know what.... I've just accepted the "broodlessness" axiom and hadn't really taken the time to think it through. Thank you for that.
1) In a supercedure, there is often a period where both the queen and her daughter(s) are laying. In that situation, there is little (if any) time that the colony is broodless.
2) In a swarm situation, the queen will typically reduce or stop laying a few (3 or 4?) days before she swarms. Swarming typically happens the day of or the day after queen cells are capped. It takes about 25 days from when a queen cell is capped for a queen to pupate, emerge, sexually mature, take her mating flights, and begin laying. If we say that the queen swarms (AKA... stops laying) when the first queen cell is capped and consider that it takes 21 to 24 days for all the brood to emerge, then the totally broodless period for the hive that a queen has swarmed from is minimal. In a swarm, the queen starts laying in 2 or 3 days and it's about 10 days before you see any capped brood.
This makes my brain hurt so I'm hoping others will chime in here on this, but I think the term "broodless" may be being misused. The key here may be in the AMOUNT of brood. In a swarm situation, there is a period of significantly less brood and this translates to less places for the mites to reproduce since they do so in capped brood.