Neither the connection between West Francia and the Ottomans and Rome is evident to me... but West Francia would have more of a claim than any Ottoman sultan.
I mean... the Ottomans destroyed the last vestige of the actual Roman Empire (it had a direct link to all the Roman emperors through history, it was the part of the empire that never fell, it did have that link) and then proclaimed themselves to be its continuation. They weren't, they came from thousands of miles to the east and took the land. Like I said earlier, this is the same as me moving to where the Aztecs lived, setting myself up with a little palace and declaring myself the last Aztec.
Francia however had a link. A weaker link than the Byzantine emperors, but a link nonetheless. Charlemagne was recognized by the Pope (who himself had a link) as the protector of the lands of the empire. The empire no longer existed in the lands Charlemagne held of course, but the Pope proclaimed him the protector nonetheless.
Not at all. Altough I don't consider that the Ottoman Empire is a continuation of the Roman Empire (for the simple fact that They did not consider themselves as Roman)
They certainly had a better claim that Frankish Kings.
First what you said about the Turks also work for the Franks. They too came from outside the Empire and settled there later. They too were foreigner that never were Roman citizens.
Sure Scandinavia may have been closer than Mongolia but infine neither were Romans.
As for your second claim, the Ottomans have better than this.
First Mehmet II was crowned by the Patriarch of Constantinople. Isn't that the same as being recognised by the Pope ?
No it is even better in fact because the Patriarch of Constantinople was the man who actually crowned the Roman Emperors for the last millenium not the Pope.
Plus the Pope crowned Charlemagne while there was still an emperor (empress) in power which makes Charlemagne a mere usurper according to Roman traditions, he could have been crown by the bishop of laliland that it would have been the same, the senior emperor of Rome is the ultimate authority to recognise who can become Emperor and Irene denied Charlie.
In the case of Mehmet II, there was no Emperor to recognise or deny him so the power to choose the next Emperor layed on the Senate and people of Rome which the man conviently conquered so arguably they did have much a choice just like the patriarch but to be fair he was not the first to access power that way.
So you see funnily enough Mehmet II and his descendent have the best claim to the Roman Empire of all the Barbarians that invaded the Empire. At least according to Roman traditions.
Did i also mention the fact that Mehmet II was also a direct descendent of plenty of Roman emperors and that in Europe only the Rurikids had comparable "roman lineage" as the Ottoman ?
Not bad for people that came so far from Constantinople.