I am not sure when I last saw an approved K3 on VJ, must be more that 10 years ago.
They still seem to be issued, well less that 10 a year, I do not know in what circumstances as I have never come across one.
An 1 130 takes about 15 months plus of course you then have the rest of the process which varies.
No harm trying an expedite, I recommend it, but it is discretionary. It seems they have a pre screening aspect which I assume is what you are referring to due to the sheer numbers they have to deal with.
If you want to bring your wife to the US for medical treatment that can be done as a visitor, even an approved expedite is not quick, just quicker.