Are cleaning products a bad idea even if I flush it thoroughly afterwards?
50:50 really. Last time I flushed a system (heating and then hot water, neither had been touched since their extremely poor installation, 20 years earlier) I ended up needing a new HW tank as the cleaning stuff loosened up something that was sealing a hole. So my crap heating system worked as well as it ever was going to, but it was (now) leaking into the rest of the system.
If the stuff that was loosen up was on the hot water side (a distinct possibility) you’ve got to wonder what was leaching into the hot water side of things.
Oh, when we took the tank out, there were some patches on it that were thin enough to poke a screw driver through with virtually no effort. Masses of corrosion………. so all in all, a good thing (semi-planned removal is better than 100 litres of water pissing through the living room ceiling.)