The history of electronics can be fascinating. People applying what seems like crude tech (today) to real problems in the day.
Fox Products lists 33 US patents on the label. It was certainly state of the art in its time. Reverse polarity protection, battery temperature sensing, using transistors in 1956 here, using PTC for charge control etc.
Even (late 1940's patents) using vacuum tubes for charge control, thyraton etc. and copper sulfide-magnesium diodes.
You can kinda figure out this charger's circuitry from the patents, a few:
US2456978 may explain the timers.
US3171076 using PTC for current control.
US3267347 if OP's charger had sense wire on the clip.
US3258672 possibly explains the transistor.
US3387200If the power transformer is old and fails a hi-pot test, the charger chassis is earth-grounded and the secondary-side may be as well.
So I would not expect a shock hazard.
I have trouble with antique gear, fun to fix and sometimes there is crossing a line towards not restoring it.
People here like to "bin it" and the antique radio forums are all about restoration. OP might find better support there.