We just had a small brainstorm with a colleague over this and we think we know why.
When you do a remote agent backup of an Exchange server, then it is the remote agent on the Exchange server itself that does nearly all of the communication into Exchange and this remote agent therefore has direct access to the library files that are part of Exchange Management.
However when you do an AVVI backup (without GRT) the remote agent on the media server does nearly all the work so needs the support files from the Exchange management. Note the remote agent requirement inside the VM for GRT over AVVI is purely to get metadata for the GRT information, the processing of the backup itself is still done by the media server.
Also just because you may have seen a Remote Agent Exchange backup apparently work in your environment without the Managerment Tools installed, does not mean that in some environments something won't work, hence we specify these tools as a support requirement in the document that has already been posted in this thread.