"How do these circuits behave while you power up (the schmitt trigger gets powered at the same time it is supposed to trigger)?"

No, it triggers ~100ms after power up as it was designed.

"WHY will no one use a supervisor chip ? They work because they are made for the job."

Right now I'm working with a pcb, and I can't add any other part. Although it could be useful in the future. Any part number?