I'm not going to claim that I know a good answer to this question, and I hope perhaps other people can contribute other ideas.
I think this is a hard thing to find information on (particularly to obtain the "richer" element of whole eggs), partly because many people deliberately eat egg whites alone to save on calories. But, like you, I sometimes end up with an excess of egg whites after baking or cooking something with a lot of yolks.
Slow cooking -- and most importantly, not overcooking -- seems to be a bigger issue than with whole eggs. Otherwise, they end up tough. I haven't tried it myself, but I've read that some people poach them instead (a "poach scramble") just to keep them tender and not overcooked.
But if you want to fry them as "normal" scrambled eggs, the only thing you can really do to improve them is add something else. Here are a few things I've tried that might help:
- Mix in a whole egg per 1 to 3 egg whites. It's not as rich or flavorful as all whole eggs, but it's more tolerable and has more of the standard "eggy" flavor. I don't find a 1:1 mixture to be that bad at all. Obviously the more you dilute it, the less it's like whole eggs.
- Add a dash or two of heavy cream (or other relatively rich dairy). It will add back a little tenderness and richness.
- Add more butter than you'd usually use for whole scrambled eggs. Same effect as above.
- If there are any herbs, vegetables, meats, cheese, or other ingredients you like in scrambled eggs or omelets, try adding some. That can distract from the lack of flavor in the eggs themselves.