I'm glad to hear you succeeded in replicating the historical capture of Cherbourg with only the UK, all despite the fact that you start out with 0 supplies when you load up the scenario. If you look at how supplies were delivered historically: Trough the mulberry harbour, and by a whole variety of landing craft/ships that kept going back and forth long after the initial landings, the level 1 harbour and 0 supplies on the ground is definitely not enough to reflect that. Of course, there were shortages, but not to the extent that is erroneously modelled in the game. The delivery of additional supplies along with landing forces is modelled in HOI3, you get up to 30 days of supplies for any force you land. (that's 30 days of sitting still and not fighting, both moving and fighting will use more supplies). The problem with the scenario you're playing is that the landing has already happened and, somehow, none of those supplies are there.
Aerial supply was also present historically, so it definitely fits within the idea of recreating the historical situation on the flawed in-game basis. I'd love to see an AAR where you attempt to recreate the historical post D-day allied campaigns in Hoi3, working around the little blemishes and flaws of HOI3. It's been a while since I've seen anyone attempt historical recreation in HOI3 and document it. Of course, the most important part is that you're having fun. If you have any other questions, and you don't find the answer, don't hesitate to ask.