On the older sets the pot shaft(s) were metal with knobs designed to insulate the user from them. E.g. by extending the knob through the hole in the cabinet so that the screw was inside, or by having it deeply recessed with the hole plugged. I can't recall when the first nylon-shafted pots came out but this was towards the end of the valve radio era. There was a kind of pot with with integral bakelite knob that could be presented through the cabinet of a TV as a small finger-twist or screwdriver-slot adjustment point, for functions like line and frame hold. But bakelite was too brittle to make a shaft for a full size control knob to fit on, for key functions like volume and brightness.
I think binning any kind of radio is a bit extreme although I can understand not allowing people to use it. An isolating transformer almost eliminates the risk of shock from most sets. You would need to take the back off, or with a card-backed set break through it, and contact both the chassis and some other point (HT+ or the other pole of the supply), but a top-of-head risk assessment suggests this is much less likely to happen than many other kinds of equally serious accident in the home. There is indeed a small risk of fire, but even the most precariously designed radio perhaps no more likely to burst into flames than a frying pan. I would not generally leave a vintage set on unattended, indeed there's not much point if you're not listening to it, so you would be on top of the situation in the unlikely event of a catastrophic fault. Vintage equipment that I expect or want to run for long periods or without supervision, I tend to attach a thermal fuse to the transformer (or the dropper of a universal) which both dramatically reduces the fire risk and avoids the transformer being destroyed by a simple fault such as a shorted reservoir cap.