We can only speculate, as I don't believe this was ever addressed in canon.
Any Goa'uld who was born before the discovery of Earth must have had another name, since the show established on more than one occasion that the Goa'uld assumed the identities of existing gods, rather than creating new ones. See this question for more on that.
Goa'uld born after this point, though, needn't have. Take Heru-ur, for example. Heru-ur is the son of Ra and Hathor, which means he may have been born after Earth was enslaved by Ra. If so, there would be no need for him to have two names; Ra could give him any name he wanted, present him to the Egyptians, and he would have been accepted as a new god by the Egyptians without having to pick one to impersonate1.
We actually see this happen on-screen in the season 1 finale, "Within the Serpent's Grasp", when Apophis announces his son Klorel (Skaara) to the assembled Jaffa. They treat him exactly as they'd treat Apophis himself, i.e. as if he were a god.
In other cases, like Apophis (Who, as Ra's brother, possibly had a name before "Apophis"), remember that most Goa'uld believed their own hype to a hilarious degree. It's entirely possible that extended exposure to the sarcophagus would drive a Goa'uld so batty that they'd completely take to their new identity. We see this happening to a limited extent in the season 2 episode "Need", with Cyrus the Godslayer. He acts like your standard Goa'uld in many respects, trading off on his reputation to enslave his people. If he had a longer natural lifespan, you can imagine how that would progress.
There are a lot of reasons why this doesn't make sense, as you've pointed out. The only way to make this logically consistent is to assume that the Asgard were the original source of Norse mythology, rather than co-opting it.
Narratively, this fits more with Asgardian nature. They're a civilization of inventors, so technologically evolved that they need Carter's help to think dumber. They create for themselves, while the Goa'uld are scavengers who adapt what others have left behind.
Why do the Goa'uld still dress like that?
For one thing, the Goa'uld are not very good about creating things for their own use. On many occasions they're described as scavengers and technological parasites, who steal technology from other races. The only Goa'uld inventor we're ever introduced to is Nerus, an underling of Ba'al who is shown taking credit for inventing the cupcake, of all things. To say that this is a race prone to stagnation is a vast understatement.
But on a more practical level, the smarter Goa'uld realize that they can only control the Jaffa as long as the Jaffa believe that they are gods. To that end, it's in their best interests to keep their culture as stagnated as possible: keep the Jaffa thinking of the Goa'uld as unchanging constants. This of course begs the question of how Goa'uld society developed to the point it was at in the show, with hand devices and hyperdrive and genetically-engineered Jaffa soldiers/incubators, but that's the point where I think you just need to suspend disbelief.
1 It's possible that Ra named his son after the mythological son of the "Ra" deity, but this seems unlikely. Firstly Horus was not the son of Ra in mythology, but was rather the son of Osiris and Isis. For another, it seems kind of silly to think that the Goa'uld would completely design their civilization to coincide with Earth mythologies.