Yet another Swiss Pass question

If your children are under 16 they will travel for free on a Family Card which you can get ( for free) with either a Swiss Pass or Half-fare Card.

The value of just being able to board transport without buying tickets first is up to you. We like it but probably not $100 worth.

From Bern to Mürren ( or Wengen) you can buy one ticket that covers the whole journey at the Bern station.

Congratulations if you have priced out all the side trips! I assume that includes a couple of mountain peaks. I do consider your estimated extra $100 to be worthwhile for its convenience and "little extras" that could easily eat up that savings. In Luzern, the recommended Swiss Transport Museum is normally 30 CHF per adult, 22 for kids 6-16, but half price with the pass. Recommended Bern museums for Einstein and artist Paul Klee cost 18-20 CHF each, for adults, 8-10 for kids, but fully covered by the pass. With a pass, you could see just as much as you like in a museum, versus buying a separate ticket and then feeling like you have to stay longer to get your money's worth.

Both the Swiss Pass and the ½ price card include urban transport (buses and trams). Of course the ½ price card gets you half price urban transport. But the central tourist areas in both Bern and Luzern are probably too small to need you to use them. See the map here for what is covered: https://www.sbb.ch/content/dam/infrastruktur/trafimage/karten/karte-sts-pass-gueltigkeit.pdf

You can buy tickets at any Swiss station to any other Swiss station, even if it involves different rail companies. Ticket machines are faster than waiting at the manned ticket desk. With the ½ price card you select the "child / ½ price" option. Tickets are valid on any train on the date shown on the ticket.

not having to buy tickets everywhere. Or is that not such a big deal? For example can we buy tickets in Bern for the whole journey to Murren?

If you are going get Swiss Travel Passes and a Family Card, you don't have to buy tickets. Just board the trains and go, no ticket required. Show your passes and card when asked. It includes train to Lauterbrunnen, lift to Gruetschalp, train to Muerren, or bus to Stechelberg and cable car to Muerren.

For lifts above Muerren, it even now includes the ride up to Piz Gloria (the top) on the Schilthornbahn cable car.
Going above Wengen, you only get a 25% discount. The Half Fare Card gives a 50% discount.

When we were in Switzerland for a week in 2016, I priced it out as best I could, and for us, the half-price card seemed to be a slightly better deal. We had no desire to visit any museums, so that wasn't a consideration. It was easy and quick to buy tickets, as needed. For me, trying to decide which one to chose was the most time consuming and confusing part of the trip planning.

The Swiss Pass now covers the entire journey "above Murren" at 100% all the way to the Schilthorn.

For me, as a single adult, the museum pass was a big part of why I got the Swiss Travel Pass instead of the Half Fare Card. Swiss museums are as expensive as everything else in the country, so it's easy to spend a lot if you are going to see more than a few. On the other hand, you probably aren't going to that many (I was in Basel and Winterthur as well as Bern). However, do look at the Ballenberg Open Air Museum, particularly if you hit bad weather in Mürren.

Yes, it was nice not to have to buy tickets every time, but I agree, it's probably not worth $100. When buying tickets, you can usually buy tickets for the whole journey at one time; you certainly can for Bern to Mürren. If you will have mobile data, you could buy tickets on your phone. If you won't have mobile data, there are ticket machines as well as ticket windows.

In Bern and Luzern, you will probably take only a few buses or trams, unless your hotel is not central (mine in Bern was not, so I needed a tram to get into the center). The centers of these towns are small and largely walkable. However, for Bern (don't know for Luzern), with a hotel stay you get a card good for local transit for the duration of your stay -so you don't need a Swiss Transit Pass just for that.

As Chris says, if you have a Half Fare Card, any transit ticket in the entire country is half off, whether it's a local bus fare or the Jungfraujoch. Just choose the "Child/Half Fare" option if buying online or at a machine. If your tickets are checked, you show both the ticket and the Half Fare Card to prove you're not breaking the rules.