Calling NeXT failed seems misinformed. As far as I understood it, Apple bought them but it was more like a reverse-takeover, NeXT's CEO Jobs became Apple's CEO, and the OS NeXT was developing basically became Mac OS X.

Indeed classes from Apple's Objective-C framework begin with NS (e.g. NSString) because they came from NeXTSTEP...

And IMO "Powered by Noyce & Moore" would sound more bad-ass than "Intel Inside".