YEah heater cores are pretty common. Why the dealer. Plenty of good independent shops around.
What I would do however is to disconnect the two heater core hoses at the firewall.
USe a garden hose and run water into each inlet back and forth to get some stuff out. Then I would fill the heater core with straight flush. I use a couple brass nipples and more hose to raise the level higher and get good internal submersion. LEt it set for 3-4 hours and then do the flushing back and forth again.
Reattach the hoses, fill the system and try it out.