Just to be pedantic (I think it's ok given the venue...)
- All an isolation transformer (which is an external component you have to supply; it's never part of the radio) does is make the mains AC voltage inside the radio "float" in contrast with the voltage coming from the wall outlet (which is of course referenced to the actual ground outside your house). This makes things a little safer because if the chassis becomes live for any reason then there won't be a potential difference between the chassis and the ground that you're standing on, meaning you won't suddenly find yourself forming part of the circuit if you touch the chassis. In these old radios it's super easy for the chassis to become live so this is a good precaution.
- Isolation transformers do nothing for safety concerning the dangerously high DC voltages which the tubes run on. In the amp itself, the AC voltage comes in, hits a power transformer inside the amp (to get nice high voltages for the tubes), then is rectified into DC, then goes into the circuit to drive the tube plates and do all the other stuff it has to do. You can still get an almighty and quite easily fatal zap if your body provides a path between high DC voltage and the chassis (which is also the DC ground). Hence the rule, only work with one hand in a powered-on amp, because at least that way if you touch something then 350V of DC can't travel up one hand, through your heart, and back out the other arm.
- The final thing you might be referring to ("some older tube radios...") around cost savings is the archaic practice of skipping the power transformer entirely and connecting the wall AC voltage directly to the rectifier. This requires various compromises in the circuit (the DC voltage produced is quite low) and you _have_ to connect these with an isolation transformer or you're really asking for trouble - the chassis is all but guaranteed to be live. The original makers in the '40s and '50s, at least keen to be seen to try to avoid killing their customers, "solved" the problem by using plastic control knobs. Here's a good article: https://antiqueradio.org/safety.htm