I've trawled through the internet and studied forum-entries - albeit with no success.
TC, when trying to access external storage, will show the usual warning to select the external storage via a special dialog and grant access from there via 'select'.
Problem is, no such dialog will appear. The Media-Player runs Android_11, yet - as I know now - this is some crippled/leaned-down offspring version called "Android TV OS" which doesn't have that notorious "FILES"-system-apk installed at all.
TC_A tells me to request a proper "com.android.documentsUI" from the vendor - which I did but unfortunately haven't heard back from support.
So I tried generic versions of FILES found on the net - still no success. Those apk's either didn't install at all or - if they did - couldn't be opened from the system-settings-apps-and-permissions-menu. Last week however I've found a new variant of this google-FILES-apk which finally WOULD install AND COULD ALSO BE OPENED!
And yet I'm back to the start again because from within this FILES-app I can access the internal storage - but it doesn't show the external storage attached. Instead it shows some pseudo-external-storage with some strange name, which, when selected, replies that this device cannot be accessed/found.

A few posts on the net do mention that FILES is incompatible with this special Android_11-variant "TV OS" - but won't give any specifics as to why or if any other workarounds for "TV OS" might exist.
So please excuse this kind-of-off-topic thread here but I do wonder if anyone in this "trove-of-abundant-wisdom"-forum might have had similar experiences with "Android-TV-OS" or would know of any other workaround, like a "TV-OS"-specific FILES.apk. Any help would be greatly appreciated about how to invoke that special selection-process-dialog (without a working FILES-apk present) as needed by TC and other applications in order to facilitate external-storage-access also on "Android-TV-OS".