Are the two filter caps simple ceramics? If so, please replace them with proper Y class caps.
There is nothing magic about Y class capacitors---they are simply capacitors which have been verified to meet all the specifications of the "Y" classification.
A bit like the CAT standards for DMMs.
The fact that a device hasn't been checked by a particular authority to correspond with some EU standard doesn't mean it is inferior.
The capacitors are shown on the schematic as 1kV DC rated.
That is probably the working voltage,(
vW)but lets be conservative & say it is peak voltage,(
vP),that implies (from memory),a working voltage of around 750v DC.
Again,from memory,it was common to use 600
vW at 250v RMS,(354vp),so these caps should be well within ratings at 120v RMS.(170vp)
Of,course,at the moment,they are loafing along with 60v RMS (85 vp) across them.
It was common to use capacitors between chassis & both sides of the Mains input in quite a lot of US equipment.
They probably assumed that most people would not get between the chassis & a good earth,& that it was only nuisance value,anyhow.
Equipment designed for the US split phase 240v system can be a bit of a trap---as such capacitors hold the chassis at "virtual earth " potential due to the "balanced" nature of the supply,so despite the higher voltage,the chassis is not "hot".
These are compatible with the Oz/EU 230v systems,but the chassis is now at the midpoint of a capacitive voltage divider across 230v RMS.
I remember buying a Drake SSR HF radio receiver,& excitedly setting it up with a makeshift antenna & earth,only to see a spark when I connected the latter.
The 2 wire cord was very quickly replaced with a 3 wire one!
Even in this case,there was no real chance of electrocution,but the zaps would be decidedly unpleasant.
Domestic "mantel" Radios in Australia mainly used 2 wire leads for many years,but they didn't have anything connected to the Mains input wiring except the "on-off" switch & the power transformer primary,so "zaps" did not happen.
A piece of equipment which is used in a workshop environment like the OP's should definitely have an earthed chassis,however,as the likelihood of rough treatment causing an internal short between Active & chassis is greater.
As it is a fixed cord,it would be simple to fit a 3 wire one.