If you turn your key, the engine turns over, but it won't start; at least you know the starter is all right. If you've noticed a weakness in the battery, you could have an alternator problem. You can typically have that load tested for free at many auto parts stores.

If your ignition timing is off, your car will not start well; or at all. You may need to get your timing reset. Along those same lines, you mentioned the "crank position center". I'm guessing you mean the 'Crank Position Sensor'. Cleaning that off is not a bad idea, but if it looks corroded or damaged you'll want to replace it.

Now let's assume its not an electrical issue. Your problem may be fuel related. You could have a clogged fuel filter or fuel rail. Along with having the sensor and alternator checked, have your fuel pressure checked. If you are getting poor fuel pressure, your car won't start.

The diagnosis from Mitsubishi may be correct, but I would check any cheaper fix first. Even if none of the items I mention help start your car, its good preventative maintenance. You may want to call other auto repair garages for quotes in replacing your distributor. The dealership will almost always charge you the highest price. A local garage may be just as capable and save you hundreds of dollars.
Good luck!