I don't find the 10 mile/6 minute call very useful because a lot happens in the circuit in 6 minutes - no-one can really know where they will be when you arrive. The one thing it does is give students a few extra minutes to worry about the other aircraft that also called at 10 miles, which they can't see because there is still many miles between them. Cue several minutes of "ABC, what is your current position" calls back and forth. The old 5 mile call was probably better.
The turning base call with intentions is very useful because it gives following aircraft the opportunity to adjust their base turn for spacing if you are doing a full stop.
A turning downwind call is useful for aircraft joining the circuit that need to fit in with aircraft on downwind. It's particularly useful when you have twins flying 3 miles on upwind doing whatever twins do, then coming back downwind at 160kt.
Joining the circuit is obviously an important call.
It's one of those difficult problems - if there is no-one in the circuit you don't need the calls but there is no radio congestion. If there are multiple aircraft in the circuit the radio is more congested, but e.g. turning downwind calls are more useful.