When you open Disk menu -> Surface test function, Hard Disk Sentinel verifies if the drive CAN be tested safely at all.
If things are suspicious (for example the drive is not responding/working correctly) then yes, it may be missing from the list of drives to be tested.
What you wrote is a clear indication that the drive is generally not stable and not working correctly so I'm not surprised that the drive is not listed.
What you wrote suggests that the problem is not related to the hard disk drive (and Hard Disk Sentinel) but more related to insufficient power required for proper operation of the drive: the hard disk may not work properly at all, may suddenly "vanish" (turn off) from the system.
This is very common issue: 2.5" external USB hard disks may not receive enough power and can't always work properly.
Please verify the USB cable, connection: maybe try a different USB cable and/or different USB 3.0 slot to ensure proper operation, then the drive will surely displayed in the surface test window.
If you use Report menu -> Send test report to developer option, I can check the actual situation, the status of the drive and the connection / USB adapter and advise specifically.
Please note that the current Hard Disk Sentinel version is very old: it may be better to update to latest 6.10 version. The latest version can show additional details, for example a warning if the drive may not receive enough power for proper operation.
> EDIT: I installed trial version of HDTune pro and surface testing is avalible there.
This is not relevant

Other tools may completely ignore the above and can force the drive test, even if it can permanently damage the drive.