IIRC, they're not normally connected direct to the signal earth or case
but via a high value resistor. And those don't fail short circuit.

No but resistors do change value. High resistance ones usually
downwards.

RFI suppression using this sort of technique has been common for probably
30 years or more. The usual answer is to plug in the signal leads before
applying the power - which is good practice anyway.

"Good practise"? Err this is customer devices we are talking about....

BTW, I have just opened up my recent DVD player to have a look (UK)
and there are NO capacitors going from mains to chassis.