Yes and no. I do make sure my FAK are well stocked and current, but I try to keep the numbers of FAK down. Otherwise it's too much to keep track of and too much to keep replace.
I get that, but it wouldn't work for me. I have different kits for different places and needs, and it's also easier for me to know that the right kit is already where it belongs, rather than worry about forgetting one. If I limited myself to one first aid kit it would be suboptimal for almost any of my regular uses.
My kits are different from the American style kits. I'm from Europe and trained using more basic medical supplies (I have had basic first aid and wilderness first aid training). Far less medication and heavy trauma things.
First aid kits come in such a wide variety I'm not sure I know what an American style kit is. I wouldn't be carrying trauma gear if I wasn't trained to use it.