Dodgeman, try one post in the middle as temporary support for a while to see if it all can be moved. A timber on the floor under the post may help spread the weight and a timber or double timber such as a 4by4 fence post or landscape timber to span a few joists above would be worth a try. If that worked then sistering or placing a beam underneath may be an option.
If you don't want to buy one of those basement support poles then get you a 4by4 post or a couple 2by4s nailed together and cut off to the correct height and then start wedging it under the lowest joist to see what happens.
And finally, if the joists are broken, split, twisted, warped, or bowed to the point they won't straighten up, then I don't see why your great idea won't work. Get a 12" metal cutting blade on a reciprocating saw. Drill a starting hole, stick the blade in while holding the saw at the best angle you can and cut what nails you couldn't pull or drive on through the floor. Its a wicked cut but doable if you don't pull out too much and bend the blade to bad. Keep us posted. There are probably other ideas floating around.