Engine will definitely still crank if the CPS is bad. There is no tie-in between the CPS and the starting circuit. However, if the CPS is bad, the engine won't start. (you won't have spark or fuel to the injectors) Testing of the CPS is included for your reference....Good Luck!
CPS TESTING PROCEDURE 1991 – 2001 4.0L H.O. engines
1. Near the rear of intake manifold, disconnect sensor pigtail harness connector from main wiring harness.
2. Place an ohmmeter across terminals B and C (A-B-C looking into connector left to right with the part with the notch in the middle on the right) Ohmmeter should be set to 1K-to-1OK scale for this test.
3. The meter reading should be open (infinite resistance). Replace sensor if a low resistance is indicated.
CPS TESTING PROCECURE for 1987 – 1990 4.0 L engines
Test # 1
Get a volt/ohm meter and set it to read 0 - 500 ohms. Unplug the CPS and measure across the CPS connector's A & B leads. Your meter should show a CPS resistance of between 125 - 275 ohms. If the CPS is out of that range by much, replace it.
Test # 2
You'll need a helper for this one. Set the volt/ohm meter to read 0 - 5 AC volts or the closest AC Volts scale your meter has to this range. Measure across the CPS leads for voltage generated as your helper cranks the engine. (The engine can't fire up without the CPS connected but watch for moving parts just the same!) The meter should show .5 - .8 VAC when cranking. (That's between 1/2 and 1 volt AC.) If it's below .5vac, replace it.