Tell us about yourself!
That's nonsense in most cases. Unless boards are coated (as mentioned in the article), all you're dealing with is solder. Anyone who has dealt with electronics for more than just the last few years has been dealing with leaded solder the entire time. And are usually fine. The problems can occur with engineers who are exposed to OTHER industrial toxic chemicals in the course of their career, especially in the past with looser environmental regulations, but dealing with just solder isn't a big deal. Not difficult to do safely. While the fumes aren't great for you, they're not toxic. And modern PCBs made for sale in Europe and other ROHS locales will have lead-free solder anyway (which I refuse to use as its melting point is so high; difficult to deal with).
Unless the boards are coated or you're melting the plastic, the only fumes you should have to deal with is from the flux, which isn't toxic. Just basic solder fumes. And if the board is modern, it will be ROHS anyway, and use lead-free solder. But I (and most I know who have been in electronics for decades) never worried much about solder fumes. I'm not aware of any epidemic of technicians getting sick from just them. It's always been from exposure to OTHER industrial chemicals that are actually toxic. Just wash your hands if touching the leaded solder.
And BTW this is a really cool project. If I can part with parting out one of my little tube radios, I'd like to convert it into an internet radio. But they're all functional, so I can't bring myself to do it. Maybe I'll find one with a broken radio, and reuse its amplification circuitry and speaker(s).
You know it's the current that kills you, not voltage. That's why many of those super high voltage mad science projects won't kill you, even with massively high voltages. Mains voltage and tube voltages won't kill you either, if there's not much current draw. I've been hit with mains and tube voltages more times than I can remember since I was too young to know better, and it's been fine. But the current - it doesn't take much to stop your heart, if it happens to go across it. It'll just hurt, and some of the high voltage tube circuits can hurt a LOT. Still, a little caution is always good.