I think I'm most bothered by Genie and Djinni, though, because those are literally pronounced the same way. Maybe D&D designers pronounced Djinni as like "Jin-knee" rather than Jeanie, though? Seems pretty weird.
That is literally how I have always pronounced "Djinni". Granted, my initial exposure to the word was through D&D, so seeing it as a type of genie would have psychologically prepped me to assume it was pronounced differently.
I can't be the only one though.
But the genie language -- and it's killing me that I don't recall it -- also spoke to genies being a unified culture, even if all four-plus groups didn't always get along. Efreet would see dao as their peers (even if they look down on them) much more than they would salamanders or mephits, for instance.
Finally, from a thematic standpoint, the language of genies is great flavor. While many spellcasters would use Draconic, Infernal or Celestial in their spellcasting, an Arabian Nights-flavored conjurer doesn't have any real connection to any of those groups. But reciting their spells in the language of genies? That's got style.
You've inspired me to decide it works thusly in my multiverse:
Primordial comes in two full dialects and four partial dialects.
High Primordial = Genie. If you speak High Primordial you can understand and communicate with those who speak other dialects of Primordial.
Low Primordial = This is your "Common" version of Genie. Basically a simplified "interplanar Genie". It functions the same as High Primordial, though it lacks the breadth and elegance. Don't expect works of literature, and expect awkwardness conveying fine technical details.
Auran, Aquan, Ignan, Terran = Partial dialects. These dialects are only minimally mutually intelligible. You can probable communicate a few basic essential concepts, but you can't have a conversation with someone who speaks only another one.
The High/Low dichotomy is the same basic way I deal with Common tongues in general. They are are all actual languages (for instance, in Faerun I make Common be Chondathan). Some characters speak it natively (High) others speak the Common version (and a lot of NPCs speak neither if they aren't from a culture where it is the native tongue).
So a Genie Pact warlock would learn High Primordial, ie Genie. A genasi might know High or Low depending on their background. Most characters who pick up Primordial would gain Low/Common, but one with high Intelligence (or role-playing justification) might acquire full High Primordial.