So there's a lot of layers to unpack here. Part of this seems to be a misunderstanding of that the "Frankish Kingdom(s)" looks like vs. what modern France and Germany consist of as well as the actual history of how the various composite parts are conquered, divided etc. Admittedly it is messy so strap in! There's a fair amount of narrative here but it is necessary to help answer the deeper question you've asked.

First, to address this we need to unpack a bit of your conception about where the "Franks" come from and what exactly "Francia" is. When the Merovingians become a going concern what we're looking at is a "Frankish" realm that consists of a whole lot of sub-tribes that over time get consolidated into three big semi-stable kingdoms:

• Austrasia (the "Old" Kingdom) consists of territory in what is now modern day Western Germany (west of the Rhine), Alsace-Lorraine, and parts of Belgium. This is the heart of the "original" Frankish territories settled in the last days of the Roman Empire when the franks were a "federate" ally and, incidentally, its where the Carolingians come from (more on this later).

• Neustria (the "New" Kingdom) is effectively what we think of as modern day northern France and was a key part of Roman Gaul. This is territory taken over/conquered by the Franks shortly after the "fall" of the Empire (note: the whole Fall thing is a whole other can of worms I can't get into here). The traditional date is 486 at the battle of Soissons.

• Burgundy is conquered from the Burgundians in 534 and becomes the last of the major "kingdoms".

Now what about "Franconia" which you referred to as being the "Frank's origin"? In fact, Franconia, is not at all the "origin" of the Franks, but a territory conquered/absorbed in the 7th century on and then effectively co-opted with the name Franconia to acknowledge that it was now Frankish territory (as opposed to say, Alemanni, Swabian etc.). So, let's put that to bed! The "origin" of the Franks, at least within Western Europe post-migration, is effectively Austrasia, the "Old Kingdom," namely the Western Rhine Valley. Neustria is a "close" second. But what about "Germania" and the Eastern Rhine? Where does that fit in?

To answer that and the question of why is France and not Germany the "inheritor" of Frankish identity we need to keep exploring the geographic evolution of Francia.

Over the next several centuries under the Merovingians and then the Carolingians (who first work for and then supplant the Merovingians) "Francia" (a conglomerate term for all the territory controlled by the Frankish kings) brings under its wing basically all of modern day France and Western Germany as well as Belgium, the Netherlands and of course some extra fun bits of Italy, Iberia, and Central Europe as well. Most important for your question is Charlemagne's conquest of Saxony, the territory east of the Rhine up to the Elbe in the East. This occurs from roughly 772 until 804 and is most clearly laid out in the Royal Frankish Annals which is available in translation and is a great source for Frankish history.

Fast forward to the mid-9th century!

In 840 Louis the Pious, Charlemagne's successor, dies and leaves behind three sons: Lothar, Louis "the German," and Charles "the Bald." The last decade+ of Louis' life had been plagued with problems managing his sons, their expectations for inheritance, their frustration with their father and siblings etc. Louis weathered multiple rebellions and when he died things "popped off" pretty much immediately.

Lothar, who'd been co-crowned Emperor in 817, immediately attempts to take control of the entire realm, despite having been confined to Italy as the result of on-going revolts against his father and specific instructions on how the realm was to be divided among Louis' surviving children.

His younger brother Louis and half-brother Charles immediately form an alliance to prevent this from happening and after a long and very bloody civil war things are finally settled in 843 with the Treaty of Verdun. The outcome is as follows:

• "West Francia" (basically Neustria and some parts of Burgundy) as well as Aquitaine are granted to Charles the Bald.

• "Middle Francia" (later called Lotharingia from whence we get the "Lorraine" in Alsace-Lorraine), consisting of Austrasia (the old Frankish/Carolingian heartlands), Frisia (the Nerthelands), and Belgium and most of Burgundy goes to Lothar, along with Italy.

• "East Francia" consisting of Saxony (the territory east of the Rhine added to the realm by Charlemagne) as well as Bavaria, Swabia, and other "German" bits go to Louis who thus earns himself the moniker "the German."

When I have students look at the map of post-843 Francia what they immediately note is that these divisions fairly nicely carve Europe into areas the correspond well to modern borders. Obviously this is a broad generalization but effectively West Francia looks a lot like "France" and East Francia looks a lot like "Germany." Over time as Lotharingia gets gobbled up by the two this only becomes clearer.

So, by the late 9th century/early 10th century we have geographically gotten to a point where there are basically two competing successor realms to the Frankish Empire. The last attempt at unity, under Charles the Fat, ends in 888 and from that point on these two realms go their own separate ways never to be truly united again. Bye bye Frankish Empire.

BUT, that doesn't answer your question of why West Francia, ostensibly the "Latin" Gallic area and not East Francia, with its Germanic culture, takes up the mantle. To answer that we need to jump back in time to address another misconception and then jump forward again.

Let's talk about your statement that "the latin-speaking Gauls (Galloromans) of Gallia adopted their name which still lives on as France." Neustria, as noted, is a key area of Gaul in a way that Austrasia never was. When the Merovingians take it over we can see clearly the ways in which they work to engage with, make use of, and co-opt Roman/Gallic culture. Clovis, notably, is baptized in 508 with the express aid of the Gallic bishop at Reims. Major intellectual figures of the Merovingian world such as Venantius Fortunatus and Gregory of Tours are of Gallic descent as well and we can see the way they bring to bear their own Gallic identities in conjunction with and in service to the Franks. This is all happening in the 6th-7th century. I'm doing a real hash of this in service of brevity and indeed the process by which Gauls and Franks intertwine and transform into a singular "identity" over time is very complex and very interesting in its own right.

Nevertheless, I point this out to give you a sense of the time-frame involved in the "evolution" of Frankish identity. By the time "West Francia" appears in 843 it has been centuries since there has been a meaningful "Gallic" identity as separate from "Frankish" identity. Latinity is now inherently associated with being Frankish. As is Christianity. The Carolingians, in fact, see themselves as the heirs to Rome quite literally, especially post 800 when Charlemagne becomes Emperor and even more so under Louis who consistently styles himself as emperor.

On the other hand, "Germany," that is to say "East Francia" across the Rhine is really only quite recently part of the Frankish World by 843. Yes there is a cultural Germanic link, but it is one that is, if anything, off-putting to the Christian Franks as they work to conquer and integrate the Saxons. One of their key goals is to Christianize the Saxons, and fundamentally the Saxons are a separate Germanic group. The Franks, are no more from "Germany" in this sense then they are from "Italy" even though the Lombards are also a Germanic people who have been integrated into the Carolingian Empire.

Talking from a perspective of Frankish origins, they were actually rather Roman. They did come from the Rhine valley, and were undoubtedly seen as barbarians by Romans, but they had close links. Such proximity to Roman territory meant trade and interaction, and we hear of Franks in Roman history, like the Frankish prisoners who escaped on stolen Roman ships in the third century and ‘plundered from the Black Sea right to Greece and Asia’ (Panegyrici Latini 8.18.3, tr. Nixon and Rodgers, In Praise of Later Roman Emperors). So the Franks came from an area which would probably have crossed the modern borders of both France and Germany. Early Frankish expansion shows why France may have ‘adopted Frankish identity’, at least more so than Germany.

As the Empire began to collapse in the 5th century, the Franks were able to expand into Gaul, and suddenly encountered a crippling problem of administration. How does a large tribe establish their rule over a vast area of land which had been Roman for over 500 years? The Franks used systems that were still partly in place from Roman rule, which was probably a major factor in their early success, but we won’t get into that. Frankish kings used their adult children and their trusted advisors to act as quasi-governors who controlled large areas of land, these are those whom Gregory of Tours described as Dukes and Counts of Francia. As well as this, (from Clovis’ adoption of Christianity) the Church was extremely important in the role of the political elite, and were hugely influential in Frankish politics. These were Roman systems and the inhabitants of (formerly Roman) Gaul would have (probably) felt comfortable being underneath them. With that in mind, you can see how the Franks were able to establish themselves as the ruling class of Gaul, and of the area that is now France, evidently out of necessity than an actual preference for Gaul over ‘Germania’.

There was little reason for the Franks to branch back towards ‘Germania’ other than purely for conquest (which they occasionally did) because Roman Gaul was a wealthy place, and barbarian land beyond the Rhine was... well, less wealthy. This means that from the 5th century onwards, the Franks would have associated themselves with Gaul, and Gallo-Romans would have associated themselves with Francia. Though Frankish origins in the Rhine valley might suggest they were associated with ‘Germania’ as well as Gaul, at that time they were not nearly as big and powerful as they quickly became at the fall of the Roman Empire, and so were probably not as nostalgic about their roots as we might imagine.

However, this does not mean that Germany simply relinquished its Frankish roots to France. Someone with expertise in 19th century history will also be able to give you the modern context of this answer; France and Germany both used Frankish history to provide pretext to territorial disputes during the period, so it became something that both countries actually (insistently) claimed ancestry to. To show the importance of this context, Napoleon was keen to promote his Frankish roots; he was very fond of the image of the honeybee (making it his sigil upon being crowned emperor, I believe) because they had found gold bees in the tomb of Frankish king Chilperic’s grave when it was rediscovered in the mid 17th century. Frankish history is rightly influential in the origins of modern Germany, but certainly not more so than France

TL;DR: Franks expanded from the Rhine valley into wealthy Roman Gaul as the Roman Empire collapsed and felt no inclination to go gallivanting into land which falls within Germany’s modern borders, they used Roman systems of administration including the Christian Church and rules over people who had been Roman for over 500 years.