So, I just discovered that LED retrofit kits exist that solve all this. They come in 4, 5, 6 inch versions and consist of an adaptor on a length of wire that screws into the existing socket. Then the new LED light gets pressed into the existing fixture. It solves the above problems. They are available in depths under 4", so there is still space for 3 or 4 inches of insulation above it. Second, they are inherently mostly airtight. Some pass an ASTM standard that says they are in fact airtight. But even the ones that don't say they meet the std, sure look airtight to me, especially if you put a little caulk under the trim before shoving it in. They are almost all made of plastic too, so transfer of cold will be limited.
They generally run about $35, but there are ones that cost more than 2X that. Lowes has a Utilitech one that is 5 or 6", 700 lumens, equiv to about a 65W, for $25. So, if you have recessed lights in a cathedral ceiling or going into an attic space, they are definitely worth considering.