She thought the Salidar Aes Sedai used Egwene and thus sought to turn their mere tool against them. Her assessment failed and so her plan did, too. It's not like she gave Egwene a skeleton key and set her loose in the secret archives or the angreal stores. As a novice always minded by two Reds and unable to channel all she could do was talk, which would've been pointless if Elaidas control were as complete as she believed. No, the irony is that if Elaidas beliefs about her own security and the Salidar desperate weakness are valid, her treatment of Egwene is unexceptional.

The problem was not that Egwene revealed some obscure but fatal flaw in Elaidas rule, but that those flaws were so glaring that rule was doomed with or without Egwene. Indeed, had Egwene never returned to the Tower Elaida would still be a damane now. Until that happened there was no move to install Egwene, but afterward it was perfectly logical. The question would be fair and logical had Egwene toppled Elaida either directly or through the Sitters and/or other Aes Sedai, but apart from the Seanchan the only person responsible was Elaida herself.