I have thought about this as well whenever a game (like Sat. night--the 'cuse vs. Auburn) goes into overtime. You could do as you suggested and place the ball on the 40 or 50 yd. line and defense would enter into the equation. Since you start off possession in reasonable field goal range (under the current 25 yd. setup), only mistakes will (typically) keep a team from scoring.
Last year's SEVEN (!) OT game w/Arkansas and Olemiss was fun to watch, but drained the players. Olemiss's coach, Cutcliffe, thinks that the emotional and physical exhaustion contributed to losses after that game for his team. In watching that game, even though it was entertaining, the level of play (especially on the lines) dropped off significantly as players had trouble blocking and playing w/any kind of speed or power at all. I am surprised that there were no significant injuries at that point.
Regardless, you bring up a good point, place the ball on the 40 or 50 and let the defense enter into it.