RS is not the only one who writes a bit of a warning regarding this route.
https://swissfamilyfun.com/murren-northface-trail-sprutz-waterfall/ says:
The waterfall detour is more difficult with steep switchbacks through
the forest with uneven terrain. It also includes walking behind the
waterfall, which can be very wet and slippery.
The path approaching the falls is steep in parts, without a railing.
So not the best for those that have a fear or heights or very small
children. You may want to skip the waterfall if these conditions don’t
sound great to you.
Assessing the weather and conditions is wise. You can ask at the tourist info in Mürren on the day you want to hike it, they will be able to tell you the conditions of the trail for that day.
When you are there you will see that Lauterburnnen does not have a long period of direct sunlight each day. Not sure what time it will start to shine in the valley on the exact days you will be there, but as far as busyness goes, I would do a morning ride. There are a lot of tourists who walk along the farm road/path and they are just enjoying the views, not thinking about traffic. You will have to pass them very slowly to avoid accidents.
The valley is not very long, I am not sure what you mean by going deep into it - unless you are going to do some mountain biking trails up. For a longer ride going towards Interlaken is a good idea, or even down to Zweilütschinen and up to Grindelwald.