The core of it comes down to this for me.
It takes x energy to get from 0 to climb speed. If that energy comes from "the ground" via a catapult or from the engines, there is no real savings. You still have to spend the energy. The only savings may be in cost of fuel to generate that energy. However, even if the savings in fuel costs are extreme, the total energy spent getting to a cruising altitude is minor compared to the energy spent keeping an aircraft up there. The cost of maintaining the catapult, would likely out weigh the cost of the "extra" fuel needed to have the engines produce the climbing energy.
Remember catapults on air craft carriers are not used because the cost savings, there used because there is no other way to get an aircraft up to speed on that short of runway. As VTOL becomes more popular, the catapults are used less and less.
Many Navies have gone to STOBAR or STOVL systems entirely. The current trends seem to be "find me something besides a catapult to get this thing in the air", even when that is at the cost of flexibility.
So to run down your points:
Less Runway : Nope, still gotta land, and even if this were true, very few locations are so tight that runways can't be extended. It may be expensive to do so, but hell, Japan (I think) build a whole new island to hold their airport.
Fuel Savings : Maybe. If you use a steam catapult and make the steam with coal, and the cost difference per unit of work between coal and jet fuel was enough, then yes, there could be a fuel savings. However it would almost certainly get offset by maintenance costs.
Faster turn around : No! It takes time to charge that catapult. It's not instant. You cancan't just launch a second aircraft as soon as the first one clears the runway. The catapult has to be adjusted, charged, and then fired. In military operations, you can only launch X number of craft. So it's possible that a military catapult could be charged for the entire launch. An airport, however is continuous. So there will be time when charging is needed. There are ways around this, like "side loading" from two sources, so that as one is depleted, the other is charging. But this would increase cost and complexity even more.