Wood floors are so cold in the winter-can’t install heated floors though

I had the brilliant idea of putting in all new wood floors. Wood floors look so sleek and classy. I also hate vacuuming and cleaning spills off rugs. Wood floors in the summer are great. I only need to mop every so often and the floors look wonderful. In the winter, wood flooring is horrible. The floors are always dusty due to my heater device. Since my heater system is on and running, dust gets into the air quality. The dust then settles on the floors for me to clean every week. The wood floors are also super cold. The cold air leaks into the house and the wood feels even colder. I can’t be barefoot in the winter. I have to wear socks and usually I put on slippers too. The heating system does not do anything to help the cold floor issue either. All the heated air rises to my ceiling, the flooring never gets touched. I have looked into this and there is really nothing I can do. What I could do is remove my furnace and get heated flooring installed. The heating would stay at the flooring level and easily warm the wood floors. The wood actually would help the heater feel warmer. I can’t do this though. In order to have heated floors, the existing floors have to be ripped up. I just bought the wood flooring, I can’t destroy it in the heating installation process. I can’t stand having cold feet all of the time. So I am thinking about putting a rug over my brand new floors due to the cold.