The answer is a little complex, but I think the main driver of the loss of his personal name is the way our ancient sources referred to him (or to each other in general).

First, a little background info on the Roman praenomen. There was, by our standards, very little variation in Roman personal names. In fact, there were only a few dozen or so in common use and fewer than a hundred are even attested (at least among the social groups for which we have recorded names). Gaius, in particular, was one of the most common names: it even featured in traditional Roman wedding vows as an "everyman" type name, a little like "Jack" in early modern English. Because there were so few praenomen, they could often be unambiguously abbreviated with a single letter (Gaius was abbreviated G. or C., Lucius was L., etc.). Families often continued to use the same praenomen generation after generation, the eldest son typically sharing his father's praenomen.

With that context, you can see how those cultural realities might lead to some issues in unambiguously identifying individuals (is the Marcus Porcius Cato under discussion the one who lived during the Punic wars? Or the one who was Caesar's contemporary?). The ancients typically solved this by noting offices the individual held or other achievements as an additional cognomen (the Cato who was alive during the Punic wars held the office of Censor, so is sometimes called Cato the Censor), or they'd use comparative titles like "The Elder" or "The Younger" (the Cato who was alive during the Punic wars was alive first, and hence is sometimes called Cato the Elder). Because of their limited identification value, the ancient sources also frequently (but by no means always) omitted the praenomen when referring to individuals. Praenomen were most useful to distinguish between multiple members of the same family who were prominent at the same time (for example, if you want to talk about the famous Caesar's cousin, Lucius Julius Caesar cos. 64BC, you need the "Lucius").

So, to return to your question, there were several notable Romans named Gaius Julius Caesar (including the famous Julius Caesar's father and grandfather). The ancients called the man we call Julius Caesar various things at various times (the dictator, the divine, etc.) to distinguish him from his illustrious ancestors (and descendants). But because he is (far and away) the most famous member of the Julian family, he was often "the" Julius Caesar, and any other Julius Caesar needed some additional explanation to identify them. Suetonius in a book titled "the lives of Caesars" identified him as "the divine Julius" to distinguish him from the many Caesars who followed (notwithstanding that several were also Julii, at least in name). However, Plutarch in his biography generally just called him "Caesar," because there was little risk of confusion.

I should also note that Caesar is not unique in being identified without his praenomen, there's a similar pattern in the way we refer to other famous Romans who we generally call by their gens names or their cognomen, and very few of whom are now known by their praenomen:

Cicero was a Marcus; Scipio Africanus was a Publius; Augustus was a Gaius; Pompey was a Gnaeus; Crassus was a Marcus; Brutus was a Marcus. The trend of eliding the praenomen is not universal though: Marcus Aurelius is a notable exception.

So that's most likely why we do not use Gaius when speaking about the famous Caesar. I can't speak definitively as to why the "Julius Caesar" format has specifically become the most common way to identify him in the popular imagination (although he is sometimes still just called "Caesar", or "Caesar the dictator"), but I suspect Shakespeare's decision to title his play "Julius Caesar" had something to do with it (as an aside, Shakespeare was working from an early English translation of Plutarch). I'll concede that this is informed speculation outside my primary area, so hopefully someone with better early modern background might weigh in.