Agreed, aircraft are extremely rare to be flying. As an example, my 2.5 year weekly campaign has seen 1 ultralight, 5 helicopters (not all at once but overall), and a distant transport plane heading for america from just before task force 34 left europe for america. Resources for the game have a small number of planes listed, and there is that A10 encounter card.
Musuem/private collection planes certainly could be put into service, just keep in mind fuel scarcity/maintenance/nuked airports and having an aircraft would tend to be high value target to acquire for anyone able to do so (plus owners needing to guard plane and landing strip). So you either have a large faction owning the craft or some sort of Airwolf situation of the owners trying to hide the location of their base of operations.
If your setting is in Sweden, keep in mind that Sweden had (has) all kinds of hidden airbases all over the country right next to long stretches of straight country road that works as runways.
Hiding in one of those with an antique plane would be plausible, since they're already kind of hidden.
EDIT: Here's a Swedish article about it:
https://www.vibilagare.se/reportage/de- ... garna-igen