NEVER put tap water in your cooling system for any reason. Distilled only.
Flush dirt from the radiator fins from tha back side it means you have to remove the radiator, but that will let you do a good clean and flush of all the hoses and other stuff in the laundry sinks. A gallon of white vinegar for a couple bucks will do the job. These bikes are old enough that a good cooling system cleaning is probably overdue.
To clean the fins just put the radidiator in a pan of dish detergent and water overnight. Next day put it face down and pour water from a bucket through it from a height of about two feet. Do that until the fins look nice and clean. To clean the innards, tape up all but one hose connection and fill the radiator with the white vinegar and let it sit for a half hour and dump it out. Then run distilled water through it until it comes out clean. That should only take about gallon. Same vinegar soak for the hoses.
Distilled water doesn't have to cost a lot. If you have a dehumidifier you have an unlimited supply of distilled water for free.
You've been into the pump, that's the hard part, might as well do a good job and pull the radiator too.