Solomon is passed over as an attacker because Solomon's Ring--her main damage ability--doesn't get the full benefit from eidolon elemental bonuses, which is an important source of damage. It's very easy to achieve an 80% eidolon elemental bonus for your team element, so Solomon's Ring and thus Solomon herself lags behind in damage once you acquire SSR eidolons with elemental bonuses that match your team element (first 3 events). One workaround is to use eidolons that provide a character-based elemental bonus instead (which fully stacks with Solomon's Ring), but that lowers the damage output of your entire team, especially at higher weapon skill levels, and thus isn't a good trade-off.