I see how majority judgement could be seen as a subset of range or score voting.

A crucial difference between range/score voting and majority judgement is that one uses numbers and the others judgements. A majority judgement ballot could list all the possible candidates or options, and for each of them, there'd be a list of possible judgements. You can say that you consider a candidate "terrible", "bad", "meh", "good", "amazing".

The idea is that humans tend to think in terms of judgements more readily than with numbers. A good ballot would find what words evoke useful judgements for candidates, as each group of voters has its own social language.

For example, with my partner we have a list of movies that we vote on. We have judgements that include "I'll leave the house if you play that sh*t", or "Omg yes!". It's great to add a movie to the list and find that one of the judgements in our made up ballot matches our personal judgements so well!

This is something I think majority judgement can do better than range/score voting: it can reflect human judgements better than with scores. In that way, it is more intuitive than range/score voting.

One benefit of majority judgement is that leaders chosen through it would know the judgement that they came into power with. If someone is elected into a powerful role knowing that half of the voters think they're "ideal" for the job, that's quite different than knowing that they were elected with half the voters thinking they were "inadequate". This means, ideally, that the legitimacy of incompetent leaders can be reduced.

Note that the amount of possible judgements in a ballot can vary. To make things quick and easy, I've had silly elections with three judgements, such as "nope", "ok", "omg yes". I've also had elections with nine judgements.

If you want to reduce the probability of having multiple winners, more judgements are a good idea. In general, the amount of judgements should depend on what the stakes are (higher stakes should go beyond just a couple of judgements), how many options there are (few options require few judgements), and the amount of voters there are (few voters require many judgements).

I think the reason for using the median is so that a judgement can be chosen as representative of each candidate. In the "nope", "ok", "omg yes" example above, if the median of the winning candidate is 3, you can tell the candidate that the score that they were chosen with was "omg yes". If the average of the winning candidate is 2.4, you can't really translate that as succintly, given that 2.4 is between "ok" and "omg yes".

I hope it's clearer why I love this voting method!