How to troubleshoot low-voltage home lighting system from 50s
Transformers don't generally go bad, and if they do they tend to burn up.
The capacitor would be the suspicious thing to me. Perhaps you can read the markings on it.  You may have to rotate it to see the markings. It's probably filtering the DC. Also, it may not be, but from the photo the &quot;diode&quot; appears more to be a full-wave bridge rectifier.
Electrolytic capacitors do tend to dry up and lose capacitance over time, and 60 or 70 years is sufficient time even for a high-quality capacitor to lose its mojo.
If you lack the ability or equipment to measure the capacitor or ripple voltage you could simply procure another of similar (or higher) capacitance and similar (or higher) voltage rating and connect it in parallel with the existing one (carefully observing polarity, to avoid a great deal of unnecessary drama).