Just to make sure I understand, are you saying it is bad for the radio, or the Amp?
I have ran several Tube Amps with Transistor radios, and never had a problem as far as the radio is concerned. If you are driving a lot of input into the amp, I would close the relay first...a lot of the old tube amps do not use vacuum relays and I would not want a load on the contacts when they close, and I would UN-key the radio/driver before I open the foot switch, this increases the chance of pitting the contacts.
I'll buy this. You can key a transistor radio right through a tube box, or you can key it all up at once. I'd also be aware of "pitting" on the relay contactors, but there is no other "detriment", or catastrophic failure that I'm aware of.