They were only able to receive signals from the bare minimum to achieve a solution (4 GPS and 1 Galileo). Their achieved accuracy was +/- 1.5km and +/- 2m/s. That is good enough in astronomic scales to get you to a planet, but it isn’t going to help failed landings or autonomous landings.
I don’t think there was any new tech involved, just a receiver put on a moon lander to see if it could detect signals. And this won’t really do anything for Mars for two reasons: 1) the signal strength would be too small for any reasonable antenna to detect GPS L1/L5 at Mars distances, and 2) the distance would make the geometry be unusable to trilaterate a solution… think about a triangle where two lengths are 100 million miles and the third length is 100 miles. That is a completely worthless geometry for trilateration of a position solution. Even if we could somehow detect a GPS signal at Mars, best case is we get atomic clock time.
Better than ballistic dead reckoning, yes. I’m not sure whether it is better or worse than star trackers plus inertial navigation units at that time scale (INUs drift over time and need to be recalibrated every so often to fix that drift, but I really don’t know how accurate star trackers are for position since I only use them for attitude measurement).