[Key Vault] Use the CAE feature.

[sadasant]

Once CAE is merged to the new core, Key Vault should use it.

Let's sync on when we get there. The changes should be small, I have already done a similar set of changes in a previous PR (that I did when I did CAE in the old core).

The reason we want to move Key Vault to the new core is to use CAE. Here's the PR where I did that with the old core (we decided to close it since we decided to have the feature on the new core): #13388