Hopefully for most species in the Milk Way their merit in the fight against the Reapers shouldn't be the deciding factor in joying the ark (urgh!). Because if it was, then it would be humans, Turians, Krogans and maybe the Salarians.
Because the asari can't win any wars
More like the writers tried to create a stereotype for every alien race, and cared very little if it makes sense or not.
The idea that the every alien race embodies a specific human trait or mind-set (or more than one, but still limited to a few),
is incredibly arrogant and foolish, and flows from the same logic of the "Humans are Special" trope.
(the real reason is laziness to actually invent a truly unique and interesting culture)
Asari can't be good soldiers because "every asari maiden" is busy being a stripper or a merc. Right.
The "Destiny Ascension" is the biggest baddest ship in Citadel territory, and it belongs to the Asari,
I imagine that the crew is composed of trained soldiers, not trained strippers.
Asari Commandos are also on a different threat level than the poor vanilla alliance soldier with an avenger mark one.
Salarians can't be good soldiers because they look like pushovers and like to sneak around?
Flash news: Technology goes a long way in this regard. Cybernetic enhancements, power armor, mechs, robots, etc.
They might be the best spies in the galaxy, but that shouldn't make them lesser soldiers.
The point is, no one did particularly well against the Reapers, because the level of technology and number of ships was simply not high enough to counter them, no matter how crafty, tough, or strong with the "powah of friendship" you are.
Everyone sucked, even the Turians and the Krogan. The petty guerrilla war they waged from that moon was largely ineffectual and more about pride than anything else.