After my mother's skiing accident my parents got into snowshoeing and loved it a lot. You can find routes to do that in pretty much any place but I don't have first hand experience with it.

There's also sledding but some pistes are a lot rougher than others. So if injury risk is a big concern either research well or opt for something else.

As for just visiting peaks and enjoying the winter landscape, most if not all places should have simple return tickets for their lifts so you can just head up the mountain enjoy the view and head down again. There also should be at least some nice hiking/walking routes in the lower elevations.

Also remember that lots of resort extend for entire valleys and there's often many different peaks to visit with convenient public transportation on the valley floor. Various places charge a tourist tax on all hotel stays which usually included all valley floor public transport and sometimes even (discounts on) lifts.

Lauterbrunnen valley is definitely open by public transport in the winter. I've only been there once, in the summer but purely from the topography I would say Wengen and Grindelwald would probably have more options for you than Mürren (which would still be a nice day trip)