Like you say, everything is relative. Architectural coatings are extremely biased toward white. I knew that's what you were talking about the moment you said "eggshell" :)
So I suspect all the colors you are contemplating are in fact pastels or off-whites. Truly dark (Munsell 0-7 lightness) or deep (3+ chroma) colors in interiors are simply not in style, and haven't been for 50 years. Here, see how the only color fit for architectural use is the top left (8 lightness 2 chroma)?
Munsell. Lightness is 2-8 bottom-top, chroma is even: 2 4 6 8 10.
When you have a "dark" (by architectural standards; probably 5 lightness) primer like this, it's for one of two reasons: First, because the chemistry of the stuff makes it better at its job; that's the case with Rustoleum Rusty Metal Primer for instance, or the mil-spec aluminum primers (the green on airplanes). Or: Second, it's tinted for undercoating extremely deep or dark colors like for warning signs or actual black or anything on the bottom half of that chart.
If the primer is merely done for color reasons, then, consult the factory, but they may allow you to add a little dark primer to white primer to dial it in to the lightness of your topcoat. But be careful; a little dark goes a LONG way! The reason to do this is to reduce the number of coats of topcoat needed for reliable coverage.
It's a regular thing for me to use Rustoleum Rusty on bare metal, followed by Rustoleum White primer, followed by 2 coats of an off-white. The Rusty is there for rust protection, the white primer is there to reduce the number of topcoats needed.
As always, primer is cheaper than topcoat, and good and proper prep is cheaper than recovering from lazy prep.