The COMPOSE buttons are disabled in the following instances:
1) The Item is in a public folder.
2) The Item is in Junk, Outbox (not drafts), or Sync Issues.
3) The item is "secure". DRM, or S-MIME encrypted. (older versions of Outlook also do not support S-MIME signed)
4) Outlook is offline.
In the situation where Read Mode add-ins are working, but Compose does NOT it is almost always #3. Usually the user will have on the option to always DRM / S-MIME their outgoing messages.
Update with image of security settings:

If Com Add-ins load a Custom Form this can also affect add-ins:
Checking Com Add-ins:
Note that outlook actually ships with some add-ins and a lot of com-addins work fine with Web Add-ins. It's only when Com Add-ins override the default new mail form (or other forms) that they can interfere with Web Add-ins.
Some COM add-ins will NOT use Custom Form, but will access the parent folder (item.Parent) and there is currently a bug that disable's Web Add-ins when a COM Add-in or VBA script does this.
File->Options->Add-Ins-> "Go" (next to manage add-ins)
