While I make bias binding (and have made a bunch in the last week) there is nothing wrong with cross- or long-grain binding.
You can even fold it over from the back without cutting it off first. Measure (and cut) a nice consistent width first. Typically though, I prefer the french bias binding even though there really is no reason most of the time for me to make it. It is great though using a striped fabric for extra interest in the border.
It's always been a struggle for me to get rid of the "excess" from the back you need to put on a long-arm. I typically don't really want to keep the long narrow pieces if they are under 6" wide, but I usually have more than that. I've met a nice person on the board who is willing to take my narrow/small scraps and I fill up a box of cutting leftovers for her every couple of months.