I have a 1988 Cadillac Seville with no heat. Replaced the thermostat. Have not had any of the usual indicators of a failed heater core. No heat. The air that comes out is enough to defrost windshield. How can I determine if the system just needs to be flushed, if it is the water pump, or if it is the heater core? Is it possible that flushing the system could resolve this issue?

I have a 1988 Cadillac Seville with no heat. Replaced the thermostat. Have not had any of the usual indicators of a failed heater core. No heat. The air that comes out is enough to defrost windshield. How can I determine if the system just needs to be flushed, if it is the water pump, or if it is the heater core? Is it possible that flushing the system could resolve this issue?

What you have is classic heater core/hose congestion, typical of Cadillac's from that era.
It is likely that both the heater core and the heater hoses that provide hot engine coolant to the heater core have become clogged and will require replacement.
Flushing usually isn't effective as only the places where coolant is still flowing will be flushed, not the area where coolant isn't flowing.
Flushing in this situation is for preventative maintanance, not repairing.

This link may help:

https://www.askmehelpdesk.com/cars-trucks/faq-how-troubleshoot-repair-maintain-hondas-selected-other-vehicles-46563-14.html#post2651405