They wrap around like the real deal, so they represent the rotation space in a linear even 1:1 mapping. Making arithmelic calculation easier. See the shortest distance on a spheres surface is not a line, rather some arc. But in the quat space it IS a line of sorts ableit a 4 dimensional one (but the formula of lines is the same as for a 3d or 2d line).
This makes interpolating stuff much easier, which is exactly what eulers do NOT do well. Since theres no singularity in the space theres also no reason why it would ever lock. Think of it as a weird mathematical way to make the sphere a plane thats still has all the properties of a sphere (which is to say its neither).
But yes the problem is much the same as why you use matrices from translate, scale, and rotate values individually too. Because the compound is not very human readable at times. But as here theres no reason why you couldnt do it indirectly by injecting values you understand.