I've got to go with Megan on this. EICAS alert you when limits are reached and perhaps it better to do something proactive to a trend than wait to be reactive. On B737, even B757/767 there are quite a few items that you would like to know about that are not on EICAS. I went B732 - B767. We did not dump old habits just because it was a new toy. I was appalled after joining a fledgling airline who have transited from B727 (i.e. with an FE) to B757/767. I used to do an overhead scan at TOC and a glance around every so often.
One day, on a jump seat, the instructor captain berated the F/O PF for looking at the overhead panel. He said it was unnecessary because there was EICAS. Just relax. He was also one of the "follow follow follow the FD" brigade.
I wonder if the A330 Atlantic glider might have fared better if the monitoring process and been more critical and considerate. No doubt there are many examples where different outcomes could have occurred with better techniques.