Although it's probably game mechanics. It's the only thing that makes sense in terms of why the PC is let into cities as a khajiit(Before they're recognized as dragonborn) especially since they've got scripts that will stop the PC from entering a city and make them get questioned by a guard first.
It's also the only reason I can think of why you can find single khajiit entering cities, but the groups(Caravans) are not. Argonians(Which tend to be distrusted by everyone in the series) seem to have little trouble entering cities.
Well, I went back and spoke to Ysolda. She said that Khajiit were not trusted because they had the reputation of being thieves and skooma smugglers and for this reason they were not allowed within City "walls." Although the conversation started off discussing caravans, she seemed to talking about Khajiit the race, when she said they were not allowed in the "walls",
Your theory is plausible and there is a lot of support for it, but like many things in TES lore, there are contradictions and support for another view. Sure, there is some inconsistency about the PC being allowed to enter cities and the fact that the PC is the Dragonborn is not a perfect explanation since the PC can go to other cities before going to Whiterun and be allowed in, although typically only after they bribe or pursuade a guard. Although most players will go to Whiterun before other cities, where they are let in because they are an eyewitness to a Dragon attack and then help the jarl, discover they are Dragonborn, etc.
City "walls" are a protective barrier. The Khajiit in Winterhold is not living inside City "walls" because Winterhold has no walls. The Khajiit running the lighthouse is allowed to keep his lonely job well outside the City walls and away from most folk. Then we have the assassin Khajiit who enters Riften, a City with "walls" that is controlled by the thieves guild. The guards in that City are corrupt as evidenced by your initial conversation with Brynjolf. It is easy to imagine that they could be bribed to let a Khajiit enter the City for a limited period of time and limited purpose.
I am not saying that I am right and you are wrong. Far from it. All I am saying is that the information from the gameworld is not sufficient to make a definitive statement one way or the other and that there is room for differing points of view on the subject. The evidence that Khajiit as a race are not permitted inside the City walls is a lot stronger than the evidence that Talos was a Bretton, yet you still hear people defending that position. That is what is great about TES lore. It is not cut and dried.