Realistically and historically, after Otto the Great united Germany & Italy and became Emperor, the Empire from 962-1806 was a German institution, the "First Reich". There was a set pattern that lasted until 1507/8 and was replaced by another formalized pattern that prevailed until 1806:
1. A German prince had to first be elected by the other German princes on Frankish soil (meaning within the Duchy of Franconia, not in France). His election gave him the right to be crowned King of Germany at Aachen (Charlemagne's old capital). Until then, he was only "King-elect".
2. Upon being crowned at Aachen, he was now officially King of Germany, which gave him the right to be crowned Emperor by the Pope. For this reason, the title "King of Germany" was also called "King of the Romans" to stress his right to the imperial crown.
3. The German king can now go to Rome to be crowned Emperor. If he doesnt, he is
not Emperor. So Richard of Cornwall and Alfonso were
not emperors; Richard at least was crowned King of Germany and so had the right to be crowned emperor; Alfonso didnt even get that far.
4. When Maximilian I was unable to reach Rome (1507), he and the Pope agreed that election & coronation of German King & Emperor should be one and the same. From that point forward until the HREs death in 1806, election & coronation as King of Germany also made one Emperor. Now, if not in 962, the imperial and German crowns became irrevocably bound.
So despite all the intricacies of the situation, and the dubious candidatures of Richard, Alfonso, and Francis,
not once in the entire period from 962-1806 was a non-German crowned Emperor, which sort of speaks for itself.
