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The CPS is a Hall effect sensor so it will not generate a signal if the engine is not spinning.
So you will not see anything at the tach. if the engine will not spin.
The fuel system has nothing to do with the starter circuit.
You stated that the engine cranked over slower when it was hot.
That is a strong indication of high resistance in your starter circuit.
Resistance can be caused by corrosion in the wires, cables, or connections of the circuit.
That resistance increases with heat soak.
The same can be true of an issue with the starter it's self.
So if you do a proper "voltage drop test" that test will tell you where the voltage is dropping out of the circuit.
But the test has to be done "correctly" that is why I suggested looking up how to perform the test on Youtube.
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