This and the fact that you also need to get a valid airworthiness certificate from the local authorities for it to fly. Getting that without OEM support is not impossible, it's just improbable for this aircraft. We keep WWII aircraft flying with one off machined parts all the time, but those parts are not difficult to machine by a modern shop. The materials used are common and the older manufacturing techniques aren't cutting edge anymore. Those older aircraft also tend to have much more simple control mechanisms (Concorde was a primitive partial fly by wire system). You would end up scouring collectors and museums for spares, not to mention corralling certified maintenance techs to work it.
Then we get to the engines, they'll need to be rebuilt and eventually be rebladed. There are enough surplus parts to keep the J79s from the 60's going, but there were thousands of those built. The Olympus 593s were a one off just for the Concord, not a lot of surplus parts floating around. Manufacturing new blades would be incredibly cost prohibitive.
My personal belief, if they want to throw billions into it, the best they'll be able to do is static runs and taxi displays. I don't think they'll get it into the air again and certainly not carrying passengers. I just don't think they have the muster to get a full D check completed and any local authorities to authorize it.
A good write up on what it would take. Impossible, no, improbable, yes.