It is a mistake to be preoccupied with the name of the throw, because, functionally, "nage" is deceptive. Properly executed, one does not "throw." Were one to use a name that describes what one does, it is "koshi-kake." (Hip-trip). In whatever position you have your feet, you establish kuzushi (uke is forward weighted on the balls of the feet), and you position your hips at a point right above his knees. The effect should be the same as if, unawares, one walked into a tightly stretched rope at that height. In classic aikido, the hips are perpendicular to the uke. Finally, it is very important to maintain your hip height until uke goes over. If you start to stand up, or heave him on your hips, you will actually restore his balance (unless he's a dive bunny). The same is true for judo's ogoshi, btw.
Certainly nage does actually throw attacker -- because it is a part of leading him. Nage must use a complete body to execute koshinage, and particularly both legs are very important in this process. The legs have the biggest muscles in human body, so it is funny to advice not to use it. One can't even maintain initial kuzushi without using a spring capacity of legs during koshinage. As Shioda sensei well remarked, all power in aikido comes from big toe. One must push his both feet (big toes) against a ground; this will create a power to throw. This power is transmitted by ankles joins, knees and leg muscles to the hips and from the hips to attacker. Power from hips will be used to maintain kuzushi(little bit as if one use a both hands to set up a bow before send an arrow) and to end the throw. More hips will spring up in the end of throw, more powerful kuzushi will be. It is particularly true with heavy attacker (that not only has a lot of weight, but also knows how to use it against nage
This is proper body mechanics in koshinage.
Of course, in case when uke jumps by himself
over nage hips, nage can safely stay all way down, he will be kind of dead obstacle for acrobatic jumper. However if attacker knows ho to use his weight, and nage will stay down, uke simply will lay over nage hips and push him down -- impossible to maintain correct kuzushi then.
For those who want to do koshinage with feet wide -- try to throw uke that weight 150 kg . LOL. I had such pleasure few times -- one learns fast to get legs together otherwise one will injure himself.