Has the alternator been tested for correct output? Excessive AC ripple? Are all of the alternator's mounting bolts in place?
Has the battery been tested? Are the battery cables clean and tight? Don't ignore the corrosion cauliflowers growing on the battery terminals.
Is the ground strap from the left hand motor mount to the frame rail in place?
Are the grounds on the radiator core support (around the battery) clean and tight?
Are the electrical connections to the alternator clean and tight? Any bent/pushed pins at the alternator?
With the key off, measure the resistance from the alternator's case to the battery negative. Post.
With the motor running. Measure the voltage at the battery back to battery negative. Post. Repeat the measurement at the alternator's Main B+ to battery negative. Post. The difference between these two measurements should be very small. Any difference indicates there is a large resistance between the alternator's case and battery negative.
Remember: no battery = no alternator.
Bottom line. Today's cars simply can not run right without a strong battery and charging system. Don't cut corners.
2003 GT Convertible (sold & missed)
2000 GT coupe (Craigslist project. Fixed. Now my DD). Windsor to Romeo swap.