APU’s nowadays provide air as well as electrics. No hydraulic power though. There are electrically operated hydraulic pumps that do provide hydraulic power though and since the APU would provide the electricity to run them they would depend on APU power until the engines are running. Of course, if the APU craps out!
Hmm, I’m wondering if these tugs would provide electric power through an umbilical. I didn’t go to the link and read the article but it’s possible!
...Then someone pulls up in a 20 year old chevy pickup with jumper cables? :-) Or they push it down the runway with an old 707 then tell the pilot to pop the clutch in 2nd? :-)