It looks like part of the reason why this happens is that AAM hasn’t been designed to recognize that all users and roles have the special exist
capability. Admin Menu Editor uses that capability to ensure that all menu items that have already passed custom access checks will actually be visible.
While that particular issue can probably be fixed with a small change in AAM, I think trying to edit the admin menu with multiple different plugins at the same time just isn’t going to work reliably.
For example, if someone enables a menu item in one plugin and disables the same item in another plugin, there isn’t really a “right” way to automatically resolve that conflict. Similarly, if different plugins have different menu order settings, those different orders often can’t be combined in a useful way. In practice, one of the plugins is going to override the others. You can’t even choose which plugin that will be unless you can somehow convince all plugin developers to agree on some cooperative way of setting hook priorities.
In general, I would recommend using only one plugin to edit the admin menu, whether that plugin is AME, AAM, or something else.