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Sorry to hear that you have issues connecting flash drives to your phone.
Would it be possible for you to test if the same thing happens with another OTG adapter? What happens if you connect a mouse or keyboard to the phone using the same adapter?
I would suspect the OTG adapter in that case even if it works with your S3. I doubt that you would get the same issue in both your Z and Z2 or with 3 different drives.
But to make sure that there's nothing wrong with the partitioning you could try removing the partition on one of your USB drives and create a new primary partition formatted with either FAT32 or exFAT. Depending on what operating system you have you may need to use a 3:rd party software for that like MiniTool Partition Wizard or similar. Also note that doing this deletes all data on the drive so make sure you don't delete the wrong partition.
It works fine for me and i've tested several different USB-drives. I can move/copy files from the both the internal storage and my SD card to the USB drive using the preinstalled app called File Commander.
This is our stock software. On Z2 i tested with 23.0.1.A.0.167 and on Z i used 10.5.1.A.0.292.
But i've used OTG many times before so it works on previous versions as well. I often use this method to copy backups from one phone to another.
Yes, i recommend that you try this with another adapter. As it happens with two different phones and 3 drives i have a hard time seeing any other cause for the issue assuming you haven't modified the phones software in any way.