I'd like to understand better how emulsifiers prevent droplet coalescence. There must be something more they do than just lower the surface tension between the droplet and the ambient substance. I tried to look up the answer but there's a lot of qualitative semi-explanations floating around that seem to choke out any precise explanation. If anyone can give a detailed explanation, that would be great!
In general emulsifiers do not work by lowering the surface tension. They do lower the surface tension, but it is still thermodynamically favourable for droplets to coalesce.
Given the above I'm sure you've already guessed that the emulsifier creates a kinetic barrier. If you take an oil in water emulsion stabilised with a typical anionic surfactant, to make two drops coalesce you have to displace surfactant from the oil/water interface into either the water or the oil. There is an energy barrier associated with this process, and this prevents coalescence.
You can still break emulsions given enogh shear. For example oil in water emulsions are used in some hydraulic systems. The extreme shear involved, in e.g. pumps, overcomes the kinetic barrier and breaks the emulsion, and this is used to provide oil for lubrication where the shear is highest.