When I jump start the car it runs and idles fine.

AGGGHHH! NEVER pull the battery cables off with the engine running. Do that once in my shop and you get a stern lecture and warning. Do that twice and you're offered the chance to find a job in some other shop! You're lucky your charging system isn't working. If it was you could easily destroy a dozen expensive computers. I did that every year as a demonstration for my students. By removing one cable on the generator test bench, it was real easy to get the system voltage up to more than 30 volts. That will kill every computer on the car and every light bulb that is turned on.

If you removed the alternator from the engine, how did you test it? There are no testers that can run them hard enough to perform a valid load test. Getting hot suggests the unit has a bad diode. That will reduce its maximum output to one third of its rated value, and that isn't enough to meet the demands of the electrical system most of the time. Output voltage can be fine and that's what most bench testers look for. You need to start with a professional load test to see how much current the alternator can deliver, and how high the "ripple" is. 90 amps is common but there are a lot that are designed for more. If all you can get is around 30 - 40 amps, replace it. In that case you'll also find the ripple is real high. That is also indicative of a bad diode.

How did you test the alternator-DVOM or VAT machine? Inspect and test all the fusible links- Might wanna go to this link for more information: https://www.2carpros.com/articles/how-to-check-a-car-alternator