I'd head straight to Lausanne and use that as my base for nights one and two and rent the car for the drive to Chamonix from there. You'll find a lot more charm. In fact, you could base your whole stay in Lausanne and not have to move hotels at all. It would only add a little bit to your driving times.
I think day 3 your is pretty jam-packed, although if you stayed in Lausanne that would be cut out of the day's itinerary and you'd have enough time. I think Gruyeres is worth visiting. It's small but pretty and charming. Stay for lunch and eat ... Gruyere Cheese, what else? If you like hiking or even just short walks and have time, check out the Moleson just south of Gruyeres.
On Day 4, I don't see any reason not to drive to Interlaken/Grindelwald but maybe keep your schedule flexible and only go on a sunny day instead of fixing it to day 4. If you've got time, take the cross-country route from Broc through Boltigen to Interlaken, instead of taking the main highway through Bern. It will take about the same length of time but the scenery is lovely and, I think would still be pretty on a rainy, grey day. You may want to do it on the way there as the road is twisty and may not be as easy or enjoyable when you are tired on your way back.
I'd consider budgeting a little time to walk in the Lavaux Vineyards, especially if one of your days is clear (although it would still be a good option on a grey day). They are a UNESCO site and very, very pretty to walk in with stunning views of the lakes and French Alps.
Instead of driving the French side of Lake Geneva on Day 5, I'd suggest you consider a drive up the lake past Montreux to Sion and take one of the mountain roads up into Les Diablerets Mountains. I found them to be different from the mountains you find in the Berner Oberland - more rugged. I really enjoyed our time exploring this part of Switzerland and found it such an interesting contrast to the Berner Oberland. If it is rainy and you don't want to go up into the mountains you can at least get a good view of the vineyards and visit the castle at Sion.
I caution you about planning your trip only around Rick Steves' highlights. His suggestions are an excellent basis for a trip but you can miss a lot if you don't expand your research outside his books/videos/etc. I find this especially true when you get outside the major cities.