Normally a bad water pump simply leaks. Normal flow, just leaks on the ground. An extreme water pump failure could be a seizure, broke impeller, or rotted impeller, but I’ve yet to see one.

My guess is the rad is clogged, has air pockets, or maybe both now that water was lost. A ½ decent rad test is to feel it with your hand from the front while it’s running and warmed up. If you feel cold spots they are likely clogged. Remember water cools as it goes across, so one side will normally be a bit cooler, but you’re looking for a significant drop and it will normally be cold all the way across. Some rad flush might do the trick for a quick fix.
If there are air pockets then there will often be surges of flow. With the rad cap off, gushes of water out the fill hole are typical signs, even after you shut the motor off. If you can run it with the cap off and it doesn’t surge, it’s probably air free.