After repointing a fieldstone foundation, should I backfill with gravel

It is not so much about mortar as it is about the working process. You should clean out the old crumbling mortar to make place for the new one. Remove the old mortar so that all parts are firm and nothing is crumbling - you can even scratch out a bit of the old mortar. The very important part is to wash the wall properly with water so that all parts hold firm and there is no dust - otherwise the new mortar would just fall out. Than press the new mortar in the joints and fill them, make them smooth on surface.

As for the mortar the lime content should not be important for the lifetime, but you can add a bit for better workability. Generally there should be no or very little lime in a stone wall. The new mortar should consist of 1 part of portland cement and 2-3 parts of sand.

If you do it properly (well prepared surface, good mortar), the new mortar should not crumble or dust for tens of years.

A personal experience: it is a very time consuming monotonous work, but the result can be very good if done with care.

Let's not over complicate this little project. I have seen these open sump holes hundreds of times. I am absolutely sure there are no structural issues here, how long has the house stood without concrete in the hole? There was probably not a concrete floor when it was built and for many years after. If you want to simply keep stuff out and hide the dirt then let's do it simple. Dig out to about a total of 4 inches below your water pipe and valve. Mix up a small bag of Sacrete concrete mix and fill the hole to about 1 inch under the pipe. Float it smooth, you're done. There is no need to get carried away with a special base etc. The size and weight of the concrete patch is negligible as far as settling or shrinking is concerned. Get the stuff and be done with it in an hour!