I don't know anything special about this but enjoy a thought experiment in the morning.
What you're describing is basically what happens in airplanes, which turn over the cabin air every 2-3 minutes through HEPA filters. They specifically engineer these systems to move air vertically, from the ceiling to the floor, rather than horizontally down the cabin so they can turn over "zones" of air quickly (5-10 rows of seats share ventilation, I think). But airplanes are much tighter spaces than buildings and have a free source of clean pressurized air to work with. To cycle a movie theater's amount of air in a meaningful timeframe would require quite a large system and the results would be noticeably loud and breezy.
Seems like a well designed system could be somewhat beneficial but would be impractically expensive for normal businesses. A poorly designed system could actually make things worse by moving air in ways that increases exposure. And simpler measures like temperature screenings, required masks, regular disinfection, and increased distances between patrons would likely be sufficient and much more economical than redesigning and retrofitting ventilation systems. I'd guess heavier droplets settling on surfaces are more of a transmission issue than lighter droplets that stay suspended in the air long enough to be removed from the circulating air.
