With EGPWS always, always, always PULL UP straight ahead.
Only with amber alerts consider manoeuvring / adjusting the flight path.
Re situation awareness; in these circumstances the understanding which comes from awareness is missing. A pop-up display of terrain cannot guarantee that correct understanding is restored, it might only add to a false impression of awareness.
In advance, consider ‘What If’, if the nav system has drifted/failed (GPS can degrade), the terrain/obstacle database is out of date, altimeter error, etc, etc. At the moment of warning there is no such thing as a false warning, you just don’t know. If such a rarity does occur then it might only be established after the event.
Pilots who believed that they knew that a warning was false suffered a fallacy of understanding, wish think, made current events fit a previous understanding. At best these people had a fright, at worse they are not available to explain otherwise. … …
Re climb performance; in most circumstances a red level pull up warning would be preceded by an amber alert at which time the aircraft should have started to manoeuvre. The time available with a red warning should normally enable a reasonably well performing aircraft to out-climb an obstacle. There are and will be a few terrain situations where the aircraft performance will be insufficient, but crews are expected to be aware of these, and plan and monitor the flight path carefully in these circumstances.