When I add a new hard drive to my TNAS device through the empty bay, do I have to pull out the existing hard drives and insert the new one to install the TOS system for it?

It's wrong to install TOS system separately for each hard drive. If you want to add a new hard drive, please insert it into the device when the device is powered on, and then go to TOS control panel > Storage Manager >RAID, and click Create, it could be used after RAID is created. TOS system will be installed to the new hard drive automatically. It will lead to RAID rename when installing TOS system separately for each hard drive, and only one hard drive's data could be shown with all hard drives inserted . In this case, please do the following steps:
1. Backup data: power off the device, insert any one of the hard drive and power the device on to backup the data
2. Delete the RAID: go to TOS Control Panel>Storage Manager > RAID, select RAID Volume #1 and click Delete
3. Create RAID: power the device off, and insert all the hard drives, then power it on, and go to TOS Control Panel>Storage Manager > RAID, and click Create. It will work properly after the RAID is created.