I was tempted to yell that that's completely unplausible in the 19th century.
It's obvious that the 19th century is the point of interest here:
No doubt, Charl's post-WWII scenario works; and before the French revolution,
"captial" is synonymous with "royal residence", which is absurd for a free city like Aachen.
The Knighthood-Emperor-Charlemagne-Barbarossa-couteous-valor-in-shining-armor-with-battlements-on-top nostalgy was most popluar in the early 19th century, the same time as Nationalism was arising in Germany.
People certainly had a taste for a reminder of past glory in Aachen.
For us today, the Carolingians might be a bit too ambiguous for having spawned France as a state as well. But for the emotional period of German romantics, there was no question as to their nationality (which is not unjustified, given that the native language of the elder family members).
I wanted to argue that these things don't mix well. But in fact they do. Even many of the Democrats had a soft spot for the kings of old.
I wanted to object that for a place like Aachen as the former scenery of coronations, it would be weird to have a Parliament debate rather than an Emperor being inaugurated. But then, the Paulskirche did the same in Frankfurt.
I wanted to remark that Frankfurt had a long tradition as a Free City, which makes clear that a (more or less) elected council and imperial tradition do not hurt each other. But again, the same applies to Aachen.
Finally, I wanted to note that Frankfurt has been a signifcant economic center and a populous city all the way form Charlemagne's days until today - with ups and downs, but essentially steadily. This does not hold for Aachen. Not at all.
So you would need very weird circumstances to relocate the Paulskirche to Aachen ...