As for the 20 gauge, it is quite uncommon in Europe except for tiny birds which are not really huntable anymore.
A partridge, hare, pheasant or pigeon really require a 12 or 16Ga.
The 16 is still very common in EU and was actually more common than the 12 in the mid 20th century.
The 16 is normally lighter than the 12, and the French gun are generally quite a bit lighter than the German ones.

All this is a moot point as wild plains game has all but disappeared due to modern agriculture.

Best regards,
WC-