Is it impossible to move land units that are out of supply during the Land Movement Phase?
It all depends on how long they are / have been out of supply and how many movement points (MP) remian.
There is a progressive MP penalty of -1 / consecutive turn of out of supply (-3 for motorized units), but until the value is set to 0, units can at least move 1 area...
So it is a case by case situation. If you have doubts, post a screenshot or send us a save.
FYI: planes out of supply can't fly out of their bases.
Conclusion: supply is indeed critical
Just reading your reply planning to report same supply issue with planes, but see you covered it all. It is necessary to understand supply to play well.