Here's a perhaps more intuitive way to think of this issue...
The precipitate we are often worrying about is calcium carbonate.
There are individual calcium and carbonate ions present in seawater.
They are moving around and frequently bump into one another.
When a calcium ion and a carbonate ion bump together, there is some percentage chance they will stick together, and start to form a solid particle.
The more calcium ions and the more carbonate ions you have in the water, the more likely they are to collide and stick just by the increased chance of these things happening.
Once a tiny bit of solid calcium carbonate has formed, then the same idea governs how fast it grows in size. As calcium and carbonate ions randomly bump into a solid bit of calcium carbonate, there is some percentage chance they will stick. More of them in the water means it is more likely a new encounter will lead to a stuck ion and a growing crystal.
So, with that as the thought process (however simplified), we can begin to understand some of the issues.
1. Higher concentrations of either calcium or carbonate (measured by alkalinity) result in more likelihood of precipitation, and once it begins, faster crystal growth.
2. Higher pH converts bicarbonate into carbonate, increasing the amount of carbonate present.
3. There are a couple of reasons that higher temp leads to more precipitation. One is that, like pH, higher temps leads to more bicarbonate being converted into carbonate. Also, calcium carbonate is actually more soluble at lower temperatures. So as the temp is raised, the percentage change of ions sticking to each other or to a surface is decreases. Both of these make precipitation faster at higher temps.
4. With making a salt mix, the idea is to add salt slowly to the water and mix fast so that you reduce the time and extent of locally high levels of both calcium and carbonate ions present near each other in and near a pile of dissolving solid salt mix in the water.
5. Magnesium gets onto calcium carbonate crystal surfaces and tends to make it less likely that future encounters with calcium or carbonate will result in a "stuck" calcium or carbonate.
Hope that helps in understanding.
