I hope you mean by stating a “polarized power cord” you actually mean a 3 wire cord set, not just a two wire cord with a plug with one taller blade.
Even with a 2 wire polarized cord set, you still would not/should not connect the neutral to the chassis. That leaves you with caps across the transformer with no actual earth per the original schematic, effectively making it an across the line or X cap rating application.
If you want to make it safe, that means a 3 wire grounded cord. It’s proper to ground anything metal that you can touch. Then your Y rated caps actually have a proper ground.
Moral of the story: Use a three wire cord set or just don’t bother.
I’m going to make an assumption here that you’re not 100% knowledgable of mains connections. Apologies in advance if I’m preaching to the choir.
Any switching of power must be the active or hot wire, never the neutral wire.
In the US, inside our standard 120vac receptacles, we use black for the hot wire, white for neutral and the bare wire for ground. Looking at the front of the receptacle, the ground is the D shaped opening at the bottom, the larger vertical slot on the left is the neutral and the small slot on the right is the hot. You’ll find the hot is connected to the brass colored screws, the neutral to the chrome or silver screws and the ground/earth is always green.
The standard in the rest of the world is that hot is brown, neutral is blue. The earth/ground is yellow with green stripe or green with yellow stripe.
If you use an old computer cord set and cut off the computer end (female end) to permanently wire it into a piece of gear, you’ll find the color coding is brown/blue/yellow-green about 98% of the time. Sometimes the ground is green-yellow and in rare cases, the colors will be black/white/green.
Re-cording old gear is always a good idea, just double check everything with an ohmmeter before plugging a unit back into mains. The last thing you want is to have the hot connected to the chassis.