Defrost not working

Look far a leaking servo, tht's the vacuum motor. Does any of them reopen after they have moved? It sounded like you said it took 2-3 minutes, then the cold set in? Also when it turns cold check the water inlet to see if it has cooled. If there is a valve in the waterline it may be closing the water flow off!

Here are some test proceedures to follow!

See if you can disconnect the heater hoses at the core inlet, then make an adapter for your garden hose with the flush kit, hook it up to the core and flush it real good, both ways! Heater cores are not built like radiators, they are a honeycomb type flow.

All heater cores are a pain in the rear to get to, not just Toyota's.

With engine at temp and heater core connected, quickly touch the top heater core hose and then the bottom one, if both are hot, then heater core is not the problem, if one is hot and the other is warm/cold, then heater core is blocked.

See if mechanic can remove the hoses and flush the heater core, removing whatever is clogging it.

But, yes, you have to remove a good portion of dash to get to core, and this is true of most cars.

As long as you flush and refill cooling system every 2 years, heater core will last a long time.

Trying to bypass the system, and mount an external heater core would be time consuming and dangerous, because you would have to run coolant lines into cabin, this is very dangerous. If they were to burst, everyone in car could receive severe burns.

If you want to save some money, use car repair manual link in this thread or get a Haynes manual and do it yourself.
After reading manual, go to local "pick and pull" yard, find a Corolla or Geo Prizm of same year and "practice" on it.

You can take the radiator to s radiator shop and have them boil it clean. Won't cost as much as a new one.