It varies with trending volumes but they all post up expedited freight on open loadboards at one time or another. I've seen and booked from just about every expedited carrier in this forum off DAT, ITS plus also a lot from free LS, and CHR loadboards and contacts. I do solo expedited tractor trailer but I know one of our husband/wife teams that also run expedited freight when it's around. The relationships you build and the reputation you earn will be your most important assets but you have to be committed 100% to seeking out this freight.
Not sure how well that works since it seems another type of freight is now your bread and butter. I'd do one or the other and do it very well or do many things and be mediocre. Me, I exploit 53' dry van spot freight, that's my market. Now I do a lot of general freight also. I follow the money. Sometimes expedited pays really well and sometimes general freight pays better than the cream of expedite. You just have to know your lanes and be in tune with the trends. It takes time.
As far as stayin out for weeks on end really there is no excuse for that in this day and age for anyone with a TT living east of the Mississippi River should fairly easily be able to get home at least weekly and earn good profits. That 3 and 4 weeks at a time stuff is from yesteryear 1970's and 80's surprising drivers still put up with that BS any more.
At my company I run my operation as I see fit pretty much the same as any independent except without the headaches plus great fuel discounts. I book and negotiate rates every load. I go home when I want without asking permission or really even telling anyone. Although I do make it a point for them to have my truck as off duty when I take extended periods of time off. I took off July the 2nd and will be off the rest of July and likely all if August also. That's how we roll here. The operator is in full control of their own destiny. How it should be for any contractor anywhere imo. I make more profit in 6 months of working my way than I ever did full time year round with someone dispatching me or making them rich on their contract freight - with no say in what I did or when, basically an employee driver covering expenses.