Intel didn't need Core or Core 2 to "save" them. Prescott successor Tejas was quite ready when Intel just decided Pentium M is better choice. Intel would have done just fine with Pentium 4 too. No idea where this Core saved Intel -thing comes from. Maybe Tejas existence was just ignored.
Because at same time AMD messed at least two architectures, probably three and Bulldozer was very rushed one. No real competition was coming from AMD even if Intel just continued with Pentium 4 line.
While Intel was not in dire straits financially, Core 2 brought them back after a few years of AMD hitting hard. Between 2003 and 2006, the K8 was generally the better CPU. It took Conroe to dethrone it in 2006. Later came the disappointing K10, and Bulldozer in 2012.
As for Tejas, there are no known samples of it, and with a pipeline of 40–50 stages, it would have been catastrophic in terms of IPC and made even Prescott blush. Had there been no Core 2, Intel would have been in a pickle.