You're right that, in comparison of its demographic importance, and that in respect to its more systematical adoption of Roman institutional and cultural features compared to Balkans, one would have thought Gallic emperors would have been a more common occurrence.

But then, Rome wasn't a democracy of course, neither its leadership supposed to be representative of anything but the Roman state. Without considering the first emperors, picked in the same Julio-Claudian family, a look at the Year of the Four Emperors might help.
While some Gauls as Gaius Julius Vindex and Aedui supported Galba from the start and initiated the revolt, the Roman legions and the peoples of German provinces didn't acknowledged his authority, maybe less because any real political problems with who ruled at Rome than for using this appropriate time to settle some old score for the latter, probably because the dynastic prestige was something important for Roman armies as early as the Ist century AD, and the rebellion was crushed bloodily.

When Nero died, the Roman legions in Germania were in the strange position of having turned from imperial loyalists, to opponents to the emperors supported by the revoltees. They ended up proclaiming Vitellius as emperor. You might note that Vitellius wasn't a Gaul. It's true, but neither a good parts of "provincial" emperors : what was important was were stationed the armies that supported them were, and which regions they tied relations with (as in this example, Gallic peoples of Roman Germania). Vitellius eventually marches south after having received the support of most of western armies, due to Othon's own coup and marched into Rome. Regardless of what his reign looked like (depending who you're talking, it ranges from violent and incompetent to popular and not that brutal), it was the only time Rhenish legions were able to impose in Rome the emperor they proclaimed.

When anti-dynastic coups took place, what won the decision was often the military capacity of the claimant : and regional armies fought each other to decide who would gain political favour.
Overall, by the IInd century, legions were rather removed from the Rhine (passing from 7 to 5, when Danubian armies included 10 legions), which was a calm enough border without real threat and whom commanders didn't had as much capacity to win prestigious victories than their colleagues in Eastern Europe or Oriens. Not exactly dull service, and the Batavii War had long term consequences in the make up of the region (possibly up to its definitive germanization), but by Ist and IInd century standards, not really the most active or prestigious : Illyrian armies got to watch and fight against Dacians, Eastern armies against Persians, Rhine army...well, against a mix of local revolts and mutinies (and by the late IInd century, police operations against big-scale robbery) or when dealing with important campaigns, these were directly led by the emperor especially under Domitianus, or when not successful general might be moved away as what happened to Didius Julianus after his success against Germans.
As Gaul remained largely peaceful (compared, for instance, to Britain), opportunities to excel and then to possibly claim favour were rare.

While the empire replaced regional institutions with their own brand (such as the Assembly of Three Gauls at Lugdunum) and while it did pleased local elites, we're less looking at a particularist movement and more of the supra-provincial identity. Provinces themselves were more seen as a fiscal and state apparatus. Not that they were disloyal to the Empire or the Emperor : far from it, during the Principate, a stress on the imperial figure was particularly strong in Roman Gaul and well answered by emperors such as Claudius, which hastened the romanization. But what defined a political identity in Roman Gaul remained mostly the civitas, not a relatively weak Roman Gallic identity.
In comparison with the population (which could be guesstimated at ranging from 10 to 14 millions for the whole of provinces, including Roman Germania), Roman armies were less obviously economically and politically necessary to the maintain of Roman structures than in Britain or Pannonia, virtually "miltiarized" provinces.
This lesser military presence in face of a rather long border, however, is what prevented Gauls to support military ambitions as diminishing the military capacity of the limes to absorb Germanic attacks was seen as risking disaster (which wasn't far from the truth).

So for most of the IInd and half of the IIIrd century, there was no real ground of a Gallic emperor to emerge and overall opportunities of military coup were rare. Gauls themselves were rather pépères, good, quite, provincials enjoying most of pax romana : praise the emperor, pay taxes, embellish your city and business as usual. They could be senators, or obtain prestigious titles everywhere in the Empire and that was obviously enough, although Gallic families didn't seem to partake a lot in the upper Roman aristocratic network, at the difference of parts of Africa, for instance.
Arguably, the Gallic origin of Antoninus led to a significant praise of the "local boy", but apart from this provincial origin, it's not really fair to consider him a Gallic emperor : born and raised in Italy, and chosen due to his ties with the complex familial network of Nerva-Antonines, he eventually shared little with the land of his grand-father.

Situation began to deteriorate a bit in the late IInd century, with renewed German raids, but there were directly managed by Severians. Amusingly, the most "Gallic" emperor of the period might have been Caracalla (born in Lugdunum, the same city his father sacked for being in his way against an usurper) who might have sported the Gallic beard (transmitted to some other emperors) and clothes (hence his name). Situation was obviously more tensed, but overall, nothing really changed politically and even the beginning of the IIIrd century crisis and the ensuing military anarchy didn't, at first, didn't raised ambitions.

But it became obvious that the Danubian border and critically the border with Sassanians were more immediate threats to the imperial order, and imperial focus and resources quickly turned there in priority while Barbarian raids went trough the limes as if it was hot butter, raiding the provinces and taking significant loot back at home (to the point one of their boats sunk under the weight of it). For Gauls that never really had to face such large scale disaster since centuries, it was both a traumatic event which was accompanied by plagues, civilian disorder and a general sense of dread. At their eyes, the empire had one job, and wasn't fulfilling it and rather took away their taxes and resources out of where they should be used, and wasn't even able to be victorious.

Valerianus tried to use a form of collegiality, making his son Gallianus co-emperor in Gaul, seconded by a skilled native general called Postumus. Valerianus having suffered one of the most humiliating defeats in Roman history, Gallienus tried to name his own son as he was in Gaul. Giving how much distrust was gone at this point, and giving how much personal prestige mattered in military and political matters in a Roman Gaul that celebrated imperial genius and honor, it ended like it did in many occasions during the military anarchy : Rhine's troops revolted, killed imperial representatives, and proclaimed Postumus emperor in 260. And so the so-called "Gallic Empire" was born, with the support of local governors, armies and ediles.