Most current countries are actually pretty young. So most terms describing people of those countries are equally young. Also most of those new terms are coined by either comparably educated people (ship captains in earlier times, governments in latter times). More colloquial forms are more likely to evolve in everyday speech.
Looking at the wikipedia page for the names of citizens of countries some patterns emerge. First of all, the overwhelming number of words end in -an. That makes sense considering the etymology you already provided.
The ending -ese is mostly referring to islands or at least places that were most likely reached by ships (in the time the words were created). That makes also sense, due to the fact that - at least for me - originating in sounds farther away than the pertaining to of -an.
word-forming element, from Old French -eis (Modern French -ois, -ais), from Vulgar Latin, from Latin -ensem, -ensis "belonging to" or "originating in."
word-forming element meaning "pertaining to," from Latin -anus, adjective suffix, in some cases via French -ain, -en. From PIE *-no-. (-an, OED, linked as well in OP)
The ending -i is almost completely limited to countries in the middle east and stretching over Pakistan to Nepal.
However the closer you get to the UK geographically the more the names are all over the place (Dutch, French(men), Spaniard). Still there are patterns, around the Baltic Sea are some -es countrymen (Danes, Swedes, Poles), while the phonetically similar endings of Greek, Turk and Kurd are also geographically clustered.
Within the UK the names end in sh(men) and self-referencing Briton, besides the Scots of course. However you still wanted to murder them a few anthems ago - and with them probably leaving that would become a nice linguistic rule again. If it only wasn't for the Isle of Man inhabitants called Manx, which is definitely a new word I just learned, but it's not a country.
So there seems some distance involved, as well as seafaring epochs (Baltics vs ocean-crossing) and some geographic clusters of names. Either of those has one or two exceptions. Finally the names in the list are not the exclusively used names for citizens of those countries. (Thank you landlocked Nepalese.)