So for clarity, they will jump to do 10 hours for 5-7 technical interviews at $0 but won't do 10 hours of paid contracting work where they get $2,000?
I can't speak for anyone other than myself but $200 / hour is pretty damn competitive for any contract work no matter where you live. That's $192,000 a year if you put in 20 hours a week (half the normal 40 hours that most folks put in). I wouldn't classify this as emergency work either. You're getting notified of a contracting opportunity in the form of an interview.
I spent ~20 years doing contract work. If someone messaged me on Wednesday, unless I'm really busy chances are I could start working with them the upcoming Monday. Sometimes I do find myself booked for weeks or months but if someone comes to me for an oddball job where I know it's only 5-15 hours of work and they're cool with me only putting in 1-2 hours a day then I can almost always find time to do it, even when I'm at near max "comfortable" capacity (~40 hours of contracting work a week).