Many people will tell you that landscape cloth does let the roots through! I think it would work. Just don't use it for its intended use. It keeps weeds down for a year or so, then the falling leaves, mulch, and whatnot join with weed seeds to grow on top of it. Pulling that whole mess up requires serious strength! So at least for crabgrass, creeping Charlie, and dandelions the answer is yes, the roots will go through!
I don't know much about air pruning, but I'm sure someone will be along who has experience with it.