Hi Traveljet -
It is difficult to give opinions about personal preferences. Certainly, compared to Engelberg or most any place in the Jungfrau region except the actual mountain tops at Jungfraujoch and Schilthorn, Pilatus has a very limited amount of walking space and distance, but very nice views of the lake. The views from the Jungfrau and Schilthorn are alpine, and you are familiar with them. There are also some nice lake views from the trails around Schynige Platte....
The very steep cog-rail train to Pilatus from Alpnachstad is a unique event in its own right. See attached. It might be worth it just for the ride up and down
These links may help you:
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And, I've attached 2 or 3 really old images from Pilatus....sorry about the technical quality.
I've only visited Engelberg briefly, but I think 6 days there might become a bit confining for me, and Luzern is a bit of a ride from there. If you weren't already planning 7 days in the Jungfrau region, I'd suggest shifting some time to there. Since you have been there, you know how much there is to do and see (if the weather is good.)
You are certainly wise to take this position:
<<"We are the sort who just want to soak in the atmosphere instead of rushing from place to place. So spending a few hours each day on a particular activity and just chilling out a few hrs after that is normal.">>
Certainly constantly moving from place to place is tiring and time consuming. Personally, I'd pick a place for a base with a wider range of types of easily accessible activities, but, now we are really dealing with personal preferences. No question that Luzern is expensive, so it may be a poor choice for a base in that regard.
Have you considered staying on the other side of Luzern, which would give you quick access to a wider range of activities?
This thread might suggest a place:
This walk is easier to access if you are staying near Luzern or around the lake:
Hammetschwand is another nice place for an easy walk, and not far off the route between Luzern and Engelberg....but closer to Luzern.
See my draft comments in this thread and some pictures there: