TYPO3 Core

TL;DR:
An old file storage that has been deleted causes the test mail send in the install tool to break / not work.

Full description:

I upgraded my instance to v10 and tried the mail sending via install tool mail setup (in env section of install toll)

Then i got the error:

Could not deliver mail
Please verify $GLOBALS['TYPO3_CONF_VARS']['MAIL'][*] settings are valid. Error message: You are trying to use a method on the inaccessible folder "/typo3temp/_processed_/".

That was caused by my BE-Logo/Login-Image which was set in my LocalConf to an Image in my SiteExtension

I debugged and got into the following class:
\TYPO3\CMS\Backend\Resource\Processing\DeferredBackendImageProcessor

There a huge return block exists, while the last condition is the one that triggers the throwInaccessibleException within the
\TYPO3\CMS\Core\Resource\InaccessibleFolder class

The issue:
I had an old file storage in the Database next to fileadmin (which has uid 1)
This storage with uid 2 was already deleted.
It had the same path as the Image in the SiteExtension i used for the BE Images
It's processing path was set to typo3temp/_processed_

It seems that the storage which definitely has been deleted, got reactivated during the install tool call - it is not filtered if the storage is still valid.