I think that there are several factors that the Aes Sedai have in their minds before sending someone forward for testing. The first is strenght in the One Power. Even if a girl came to the tower with say Lanfear's potential they have to wait for her to develope a certain level of strenght before she can take the Accepted test simply because she needs to be able to manage a certain amount of the One Power to advance to the more complicated weaves the Accepted learn.

Second is skill. Being accepted is a bit like going to secondary school/High school. The teachers there dont expect nor at they set up to have to teach you to read and write or how to do simple multiplication tables. Its taken as read that you have that basic knowledge already. Likewise part of the Accepted training requires a basic knowledge and mastery of certain simple skills. So until a girl reaches that stage she can't go forward.

We have seen that skill often follows power so if someone has very strong potential they pick up the skills fast as well.

Next would be general education. Most people in the Wheel of Time seem to be able to read and write but I would guess there are some who can't and the standard amoung those who can varies. Likewise some from noble backgrounds may have a basic understanding of the Old Tongue and know a bit about other worlds. The candidate has to reach a certain standard before they can go on to Accepted studies and likewise the wood be Aes Sedai much have a basic general knowledge before she is let loose on the world. A wofully ignorant Aes Sedai would reflect very badly on the Tower.

Lastly there is probably the requirement that the girl has adopted to the Aes Sedai way of life. They are not all cut from the same cloth but all of them put the Tower as No 1 of their list of priorities. To pass the Accepted test the candidate has to want it more that anything else. Likewise the Tower does not want an open rebel gaining the shawl.

There is probably a bit of give and take among all these factors depending on the circumstances but gaining strenght along is not the be all and end all unless the White Tower is facing imminant desruction like say the Trollock Wars or under Hawkwing.