A smart cook I used to know said buy good meat, put only salt on the frying pan, cook to 'taste' re doneness.
I still follow that, buying good beef from the butcher with at least 15% fat in it, for the taste, and now I occasionally use sea salt instead of regular because I kind of like the larger salt granules. I prefer somewhere between medium rare to rare.
I've not ever worried about refrigerating the patties I've made, assuming I cook them right away and don't leave the meat out to spoil. Don't remember any problem with crumbling. Oh, and my patties are thick, maybe 3/4" thick.
The only time I'll use eggs or anything else is when I'm making meatloaf - which I consider a whole different animal from a burger.
The biggest problem with burgers, to me, are the buns. Regular store hamburger buns seem to taste/hold together worse these days, and I always have my eye out for a sturdier more flavorful bun or roll.