Clamping the output ground very close to earth ground
The fact that many power supplies have a floating ground is often a feature, rather than a bug!

You want to avoid ground loops and have the ability to connect multiple pieces of equipment without causing lots of current to flow through the connected grounds. Ground isolation via transformers, differential signals, optical, or RF actually is a way to avoid a number of safety and signal issues in interconnected systems. Ground is not magic. It's just a reference point, and a potentially signal-contaminating one if you get confused and think it's the same in two different places. It can be better for the designer / engineer to decide exactly where to join grounds, rather than having the power supply do it for you.

Your measurement here is also potentially flawed: as you say, "the current probably won't be that large," meaning that this is probably a very high impedance floating condition. (High voltage divided by low current equals high impedance.) The mere presence of a voltage difference probably doesn't actually tell you all that much.

If you did want to tie the input and output grounds together, I'm not sure why you'd want to use a diode rather than a wire!