The four characters
The claim rests on four characters in the Analects, 有教無類, and they are among the most disputed in the book. The usual modern reading is in teaching, no categories: teach anyone, whatever his station. The oldest commentary on record read it that way too.
In the twelfth century Zhu Xi turned it round: where there has been teaching, the categories are gone. Education dissolving the distinctions between men, rather than being offered without regard to them. James Legge called that gloss very extravagant. The open door is the older reading.
The twist
For nearly six hundred years, from 1313 until the examinations were abolished in September 1905, the Chinese state examined the men it appointed on his book, in Zhu Xi's edition. So the examinations usually held up as proof of the open door were built on the newer reading, not the old one.
Who actually got in
Even on the generous reading, the door he opened was a door into government, and it opened at the level of a man with a name and no money. Elsewhere in the Analects the common people may be made to follow a path but not to understand it. He was a defender of hierarchy who wanted the hierarchy based on something other than birth.