Flushing the radiator won’t be harmful if you do it right (and it’s pretty easy to do). You can buy a flush kit and antifreeze at any auto parts store. You’ll also need a garden hose and water supply.
What you’ll end up doing is cutting one of the heater core hoses and attaching a T fitting that you can connect to your garden hose hose. Connect the hose to a good water supply (like the hose bibb on a house, for example). Then open the petcock at the bottom of the radiator and turn on the water. The coolant will start to drain out of the radiator and onto the ground (treat that later), but as long as you’re running water from the hose, you’ll be continuously supplying cool water to the water jacket around the engine, the heater and radiator.
Just run the engine and let the water circulate and flush everything from the radiator until that water running on the ground is running clear. That lets you know that you’re done. Never run the engine if your water supply is turned off! It’s a good idea to keep the radiator cap off while you do this so that you can occasionally monitor the coolant level by sight.
Turn the engine off, close the petcock and let things cool. When the engine / radiator are cool to the touch, drain the radiator at least enough to add the correct amount of antifreeze to the system.
Don’t leave any puddles of drain water with antifreeze lying around The glycol antifreeze has a “sweet” taste that attracts animals, but is poisonous to them. Flushing and diluting it on the ground makes it relatively harmless. (This is all my recollection of the process from about 20 years ago, the last time I had to do it on an old pickup truck with about 150,000 miles on it.)