In the Age of Legends, the "sedai" was an honorific that showed they were a humble servant that served the public in a particular way. When the half-trained women who survived the breaking and the time of madness founded the Tower, they probably kept it to maintain the connection with the past. They also probably saw themselves as serving the public good by being a bastion of Light and knowledge in the dark age.

Now, though, as you point out, every sister should know the meaning of sedai. We know that the modern Aes Sedai take pride in their humility... see when they set themselves punishments, when the Amyrlin must be willing to wash the feet of the Sitters, etc. Bearing the title "servant of all" is something they are taught to hold with honor.