Use to work with a guy who's old Ford pickup would stop putting out heat every so often. When it did he would switch the hoses around and have heat again. It's the cheap/easy/lazy mans way to go.
Might sound silly but you might try switching the hoses around on the heater core (or engine side). Keep in mind that they might be different sizes, so you will have to contend with that. This will amount to flushing the core to some extent and is a cheap/quick thing to try.
careful not to push the hose on tootight as 30+ lbs is more than the core was designed for and it'll look like an exploded accordion!
Yes, don't get aggressive with it.
More food for thought: as a young kid I didn't have much money. Had a friend flush my heater core as it wasn't given up any heat. It was flushing well but still no heat, so I gave him the green light to keep cranking up the pressure. Thinking was that a no heat situation didn't do me any good no matter how mush flow the flush machine showed. After we blew the core I went in to finally replace it only to find the heater box filled with a mouse nest. No air could move across the core. Lesson learned to check everything before blowing up a perfectly good part.
From a guy that spent a cold winter with no heat in the car as a young man, best of luck with it.