Transmission fluid and coolant intermixed in transmission and cooling system. How thoroughly do I need to flush them? Recommended flush technique?

So my 2004 Dodge Neon’s transmission cooler failed. (Neons are notorious for this.) This caused transmission fluid and coolant to mix together, both in the automatic transmission and in the cooling system.

The good news: when it failed, it spilled out on the road all at once instead of subtly leaking internally over time.

The bad news: I unwisely let the car sit for 5 months before starting repairs. But at least, according to my mechanic, coolant is denser than transmission fluid, and they don’t blend (like olive oil and vinegar), so the coolant should have settled on the bottom of the transmission pan with the transmission fluid resting on top of it.

I just finished replacing the radiator. Now, I’m wondering what the best way to flush the cooling system and transmission is.

COOLING SYSTEM FLUSH:

Yesterday I flushed it using the Chilton manual technique: I disconnected the upper radiator hose, removed the thermostat, stuck my garden hose into the radiator, and ran the hose until the water ran clear.

Is this enough? Because on StackExchange and YouTube (ChrisFix), the recommended procedure is filling the system with distilled water, then running the engine a while to let it circulate, optionally with some radiator flush chemical added.

Would this be overkill? If so, I’d like to avoid it — temperatures are now just below freezing for 18 hours a day where I live, making water flushes tricky and possibly risky.

TRANSMISSION FLUSH:

Both my mechanic, and various online sources (namely, scores of negative user comments cursing transmission flushes for destroying their transmissions), have warned me away from power transmission flushes, saying the risk of transmission damage is too high.

Instead, my mechanic recommended I flush the transmission by changing the fluid twice, being sure to run the car in every gear for 5 minutes apiece after the first fluid change. (And also to change the transmission filter.) Again, he said that coolant will settle below transmission fluid (like olive oil and vinegar), so emptying the pan should dump most of the coolant out.

What’s StackExchange’s opinion of transmission flush techniques?