how to use fn_xe_file_target_read_file to read files over network
As Kin suggested in his comment, the F$ share is problematic since that only allows administrator access.  Although you've provided file-system access to the folders in question, the account does not have access to the share.  Create a share specifically for the xel files, and provide access to the SQL Server Service account on both the Share, and the folder.

Using the Universal Naming Convention name, or UNC, is preferable to accessing the file via a mapped-network-drive.  A mapped network drive just obfuscates the true location of the file, and you'll need to somehow create the mapping inside the profile for the SQL Server Service account (that's what Kin is suggesting with the net use command he shows in his comment.)

My SQL Server 2012 instance reads files across the network perfectly like this:

SELECT *
FROM sys.fn_xe_file_target_read_file(
    '\\machine\share\ErrorLogs\system_health_*.xel' --this is the UNC
    , null
    , null
    , null
    );