The biological balance is in their favor while not being in yours (or mine at the moment).
When you have stable biological conditions and a decent coral load the system is just better at resolving what would be unsightly algae and bacterial issues in an unstable or newer system.
Flow and lighting can also be contributors to cyano taking hold.
Yep, this.
In short it is about two things: competition and consumption.
Competition because corals, algae, and other organisms compete for the same nutrients - and while algae at least in the beginning is 'better' at doing it than corals, once corals reach a certain amount their consumption of nutrients in total is more than that of algae. That's one aspect.
Another aspect is consumption. In an established tank and cuc are constantly cleaning up algae, it does not give a chance for algae to spread everywhere. They will constantly pop up, but will also constantly be consumed by cuc. Although that will also generate nitrate, phosphate, and other metabolites, which is also why some aquariums have 'elevated' readings for this.
In other cases, like mine, I have a pseudo-algae scrubber that has a large variety of algae (hair, cyano, etc.) which just constantly deplete excess nutrients no matter what amount of nitrate and phosphate are produced in what proportion, no matter how much I feed (I probably feed about 80 to 100 small NLS pellets a day for a 20 gallon tank). At some point I did have various algae grow everywhere and all the time, but now has naturally slowed down despite increased feedings.