Back in the days of identical two prong plugs, if something had a metal case (as many things did back then), if the internal wiring shorted to the case, the user could get a nasty shock. So they made the neutral prong wider, since neutral was connected to earth ground. If you had something with a metal case, and something shorted to the case, now it would just blow the fuse.
But there was still a problem. If the neutral wire broke and you turned the device on, the case would no longer be at earth ground potential, and you’d have the full 120 volts on the case, limited in current by the internals of the device (the current would go though the device and then connect to the case since the case was connected to neutral, which was now broken). Again, a pretty serious shock hazard.
This evolved into our modern system, where we run a separate safety ground.
If the hot wire breaks, the device just stops working. No shock hazard.
If the neutral wire breaks, the device just stops working. No shock hazard.
If the safety ground wire breaks, the device keeps working, and still no shock hazard (though you’ve lost the safety at this point).
If the hot wire shorts to the case, it blows the breaker.
If the neutral shorts to the case, no real shock hazard, but if there is a GFCI present it will trip.
The safety ground is already connected to the case.
You can still end up with a shock hazard, but it requires more than one fault.
But what if you have a device that has a plastic case, and all of the electrical bits inside are insulated? There’s no real shock hazard no matter what shorts out or breaks. So in these cases, there’s no need for a polarized plug or a 3 prong plug. You can use the old fashioned two prong, non-polarized (both prongs the same size) plug and there’s no safety hazard.
There are rules that devices have to follow in order to use a symmetrical 2 prong plug, but as long as there is no shock hazard from them, they are safe to use.
By the way, the reason you attached to the radiator back then was that your water pipes were used as the home’s earth ground. These days, a separate copper rod (sometimes more than one) is driven into the ground to make the connection, since water pipes can be plastic and can’t be relied on any more. Metal water pipes are also connected to this copper rod, but that’s to make sure that the water pipes never become a shock hazard. Your home doesn’t rely on the water pipes any more for its earth ground.