HeliComparator, thanks for being a voice of reason and experience. I am a 3000+ hour fixed wing pilot with multiple "hairy recoveries" several on instruments in bad weather. All of them required being the "master of my aircraft".
I'm no Sullenberger, just a guy who grew up with old school pilots who taught me how to be the master of my aircraft. How you get there has been extensively discussed here and I have little to add except "cowboy flying" has its place. Let me put it this way: When I taught my kids to drive we went out on dirt roads with plenty of grass on the sides and drove faster and faster till things got squirrely around the curves. Just enough for them to recover or loop the car. We also learned to surprise brake in the shortest distance on pavement and dirt, no ABS allowed. I encouraged them to regularly go out and push the envelope within reasonable safety limits (judgement required).
I just got into rotary (yes at my advanced age!) and am flying only manual in the OH-58 and 407. I get the push button aviation problem. It has been around in fixed wing for a long time, way before glass.
My experience is missing the "keep passengers comfortable or risk losing your job, even it it means dying". If this is really common, and "flying the rails" is driven by that mentality, then it's a giant safety problem created by management holding the threat of termination or demotion over pilots' heads. I know people who, when confronted with serious s**t, will think twice about squawking 7700. Amazing, since that is the declaration of emergency they should make before even opening their mike, and it immediately removes one worry from their plate -- deviation issues or worse on instruments. They are thinking "Oh s**t the Feds are gonna pull my ticket". Ridiculous.
It is much easier to fly around on autopilot, but doing it when not really necessary (CAVU weather, etc.) is stupid unless the aircraft requires it for all ops (e.g. SAS). Fly the aircraft and do your own navigation as a rule. Then when things get really tough, you can fall back on the autopilot and create a safety and workload buffer. But some day that fresh flying skill will save your a**.