<<"Lucerne is a somewhat larger town, to Swiss standards, but near the mountains and lots to do.">>
Lake Lucerne (der Vierwaldstättersee in German) is a good match, I think. This link should provide a nice overview map.
Its nice because it is a simple map for generally relating to the area.
The city of Lucerne itself itself is very attractive, and, in many ways , quite sophisticated.
Very close to the train station is the Rosengart Collection - world class Picasso and huge Klee collection. Just one example.
Hotels tend to be expensive and full of foreign tourists in the summer, but there is certainly a wide range of places to eat and sleep.
The lake is ringed with trails, accessible by all modes of public transportation. And, it is beautiful. A ride on the lake boat from the southern end at Flüelen north to Luzern (or vice versa) is a delight, with varied scenery. Therr are a lot of towns on the rail line between Lucerne and the Gotthard Pass, and they can be considered for a base. Brunnen comes to mind, but there are others.
I like to go to Hammetschwand for lunch,by lake boat and funicular plus a nice walk. Fantastic views
The Swiss Trail (der Weg der Schweiz) goes around the lower part of the Lake. It's a really nice walk on good trails, with a few significant ups and downs to make your heart and lungs work.
There are also some more challenging trails. I'm looking at Klewenalp to Niederrickenbach, but, so far, only looking. ;-(
Canyoning is popular around the lake; it can,of course, be risky. There were some deaths in one poorly managed event a few years ago. Or, maybe it was near the Brienzersee.
There are nearby mountains, as you say. Pilatus is spectacular, Rigi is nice. Some nice walks on the Rigi.
We stay frequently at Rigi Kaltbad, and love it, but it is a little inconvenient to use for a base because the transport up to Rigi Kaltbad stops in the early evening. The pleasant walks up there are good for a couple of days, but it takes major skill with the schedules to go for a day trip to, for example, Klewenalp across the lake, and still get back up before the transport stops.
I wonder about Interlaken as a possible base in stead of Meiringen. Excessively popular with tourists, and we have never stayed there, because we always go in to Wengen or Grindelwald. But it has good access to the Lauterbrunnen Valley, Wengen, etc., Schynige Platte (nice walk up there and also a serious hike past Faulhorn to First.).
Meiringen has a bit more "Swiss" feel for me, and , is closer to the Grimsel. I like it better as a place to be; for me, Interlaken is only a transfer point. That is for me. But not for many others. Not so many hordes of tourists in Meringen as in Lucerne or Interlaken in the summer, but there will still be quite a lot.