Your 'scope is unlikely to be useful in this case.
GFirst, the TC-600 was avaialble in VHF and UHF versions, make sure they're all the same version (I'd be surprised if you could program a VHF with a UHF data file).
Can you clone between radios with a cable? If so I'd consider cloning a working radio to a non working one.
Were they new when you got them or, as I suspect they might be, were they second hand?
If they were new, return to supplier if cloning doesn't fix it.
If second hand were they meant to be working?
If they were sold as seen then check antenna connections, first swap a good antenna onto a 'faulty' radio.
If it works order new antenna.
If it doesn't then check the antenna connection, make sure it's not loose or that there's physical damage to the connector, also check antenna socket solder connections to the board