Ease of mixing and reproducing those mixes.
The darkest color in the triad is the darkest value you can mix - in fact adding one of the other colors will lighten it to some degree. It's necessary to match your triad to your subject for it's value range. Usually the blue is the darkest component, so a triad with cerulean wouldn't be ideal for a subject with strong darks.
With any triad you need the primaries or their rough equivalents to mix secondaries. Regardless which 3 colors you choose, at least one secondary will be dull; that is, not a true orange, green or purple. Often 2 will be dull and all 3 will be dull if you select a low chroma triad, like ultramarine, burnt sienna and yellow ochre. It's also necessary to match your triad to your subject for color chroma. The bias of your components will affect your mixes: A violet bias red like alizarin makes clear purples but dull oranges. Cad red makes better oranges but earthy purples. Prussian or pthalo blue bias toward green so that's their strong suit in mixes. Ultramarine has the edge with purples.
Some triad palettes with their strengths and limitations:
Ultramarine, burnt sienna and yellow ochre: Good natural earth tones and greys. Very dull orange, no real purple and your greens will be medium to low chroma. Darkest color is ultramarine + burnt sienna which can go nearly black. A very traditional palette and surprisingly versatile despite its limitations. All 3 colors lift easily if that's a requirement.
Ultramarine, alizarin or equivalent and cadmium or hansa yellow: Vibrant purples. Vibrant greens except in the blue green range. A slightly dull orange which can almost mimic burnt sienna. Strongest dark is purple-ish. Good earth tones can definitely be mixed from all 3 components but requires care.
Prussian blue, alizarin and hansa yellow: Very vibrant greens including the bluish greens. Good purples, same orange as above and similar purplish dark. Substitute pthalo blue and quinacridone rose for a highly transparent looking palette. Again great earth tones are possible but tricky to mix.