When I built my superinsulated house in 2014 with radiant heat as the only heat source, my builder insisted I get a heat recovery air recirculating system, because of the reasons the original poster mentioned, and the fact that my house is so well sealed. It has both supply and return ducts to most of the rooms. It's a single story house with a full basement, so these ducts were run in the basement ceiling.

Last I knew there was no objective data showing that domestic air filtration systems work. I have not exposed myself to that world for a few years so I might have missed something. Duct systems tend to cause pressure differentials that suck in outside air in uncontrolled ways. Best to avoid them or approach them very carefully. Depending on what sets off your asthma, it might make it worse. A well designed HRV or ERV system could help if outdoor allergens are your problem. They can be set to a slight positive pressure keeping the bad stuff outside. If indoor allergens are the problem, very good cleaning with a central vac system will probably serve you better than an air filtration system.

Interesting. I live in Colorado as well and installed a radiant heat system. I am looking at the HRV system. My house is so dusty and fine particulates build up. Why, because there is no exchange of air.