Funny since Shepard finished the ardat yakshi mission for them
That's because as the protagonist, Shepard is a special snowflake. He or she is always going to succeed where others fail, and any time Shepard runs across allies in missions it is always going to be up to Shepard, rather than those allies, to save the day. The same thing occurs on Menae where Shepard fixes the Turian comm tower, on Sur'Kesh where Shepard rescues the Krogan Cerberus is after, on Tuchanka where Shepard destroys the sole Reaper on the planet, and with the Quarians needing Shepard to both destroy the Geth dreadnought and the lone Reaper on Rannoch.
In any case one failure doesn't alter the lore that the Asari field the galaxy's best soldiers. In the real world there isn't an elite military organization in existence with a significant combat record, that hasn't also lost a couple fights.
But the Reapers avoided them. Why? There's a reason. In military history, there's always a reason why a conquering invader avoids a potential target. Usually, it's that the target provides a combination of a staunch defense and not much to offer. Whatever the exact composition thereof, the Salarians didn't look like very tempting targets for the Reapers, and so the Reapers avoided them.
There is no indication that the Reapers saved Sur'Kesh for last out of any fear of Salarian military prowess. The Reapers were qualitatively superior to every other faction in the galaxy, and possibly quantitatively superior as well. With a smaller fleet than both the Turians and the Asari, there is no reason why the Reapers should have viewed them as more of a threat.
That the other Council races were struck first might also indicate that the Reapers viewed them as less of a threat than the others. For a real world comparison, during the Second World War (apologies for the Godwin) the United States adopted a Germany first strategy. Having said that...I'm not sure why the Alliance would be viewed as more of a threat than the Turians, Asari, or Salarians (smaller fleet than each), but the series did too often venture into the humans are special trope.
Tuchanka was most definitely invaded. Maybe not by a large force, but those Brutes and Marauders and Cannibals sure weren't tourists fresh off a cruise ship. Earth and Palaven didn't have an advance warning of any real kind. A bunch of Batarians showed up; that doesn't really mean anything before the fact. Khar'shan got whacked and shut down before anyone knew what was going on. Thessia got hit after Khar'shan, after Earth, after Palaven, after Tuchanka, even after Cerby attempted a commando raid on Sur'Kesh. Didn't matter. Even with advance intelligence, they didn't adequately prepare their defenses.
I'm not sure what went down on Tuchanka can be called an invasion. A single Reaper was sent there, and a smaller one at that, with a comparative handful of troops. At best that operation was a raid and not an invasion. As far as the Reapers are concerned Tuchanka seems very much a sideshow compared to the invasions of Khar'Shan, Earth, Palaven, Irune, Dekunna, and Thessia. Further indication that Tuchanka was a relatively small scale and limited military operation by the Reapers is that after Shepard sorts things out the Krogan are free to reinforce the Turians on Palaven and take part in the counterattack on Irune, rather than being bogged down defending their own turf.