Sadly it is not as simple as that, and additionally a lot depends on local regulations. Lets think about some tiny airport out there that has only one IFR approach, lets say it is a VOR approach. No RNAV approach, no ILS, GLS or MLS.
Now, according to what we are allowed to use we would set up the VOR approach in the FMC and fly it using IAN which makes it look and act like an ILS, autopilot or not and flight director or not doesn't matter, i can fly it either way. However we do not tune the VOR as we are not required to do that, perhaps set up the approach for our alternate, or some nav aids used during the missed approach or simply forget about it all since we do not need to tune anything, we just have to make sure there is no ILS tuned, but that is simply an autoflight system restriction (approach mode would try to intercept the tuned ILS instead of the IAN approach if one is tuned).
The VOR on the ground has to be serviceable of course or ATC could not clear us for the approach. But we won't use it, except as backup, since all we need to fly the VOR approach is one FMC, and we got two of those. Now, how do we now that we are still within the limits of the approach? By using a generic RNP of 0.3 which is database coded into the approach. The approach in itself does not have any RNP, however if you want to fly it using the FMC as sole guidance you have to use an RNP and it is part of your companies approved procedures and therefore part of the AOC.
The need to tune, identify and display the navigation source for a non precision approach is not there anymore. That bit of legislation may not have reached your company or country yet, but it will. The aircraft we use are certified for it for quite some time now, the 737 since it turned into the NG.