That looks a nicely constructed converter. Plenty of glass fibre insulated sleeves etc bode well for the quality of other parts. A low voltage AC-DC SMPS would, of course be isolated between input and output. A mains AC-AC converter - I really wouldn't make that assumption.
I can understand the wish to carry through the protective earth from input to output. ESD leads (should) contain a series 1M Ohm resistor. Any high frequency noise could be imposed across this resistor, possibly damaging ESD sensitive parts. Additionally, If the Hakko, or any other equipment that you intend to plug into it, has a 3-pin plug, it must absolutely be mains earthed, no quesiton - It is not good that the 3-pin output sockets are not earthed by default.
In terms of reservation, the only one I would have is implementation. If your earth wire becomes detached from the input socket and contacts something internal, you have the worst possible scenario. Make sure your connections are mechanically secure before soldering, use heatshrink sleeving at the ends and take a leaf from the manufacturer's book and use heat resistant sleeving to protect it from live parts, heatsinks, transisor tabs, etc.