You're assuming they did this to invade Citadel space. We don't know that. Perhaps they wanted to see how well the Rachni fought a war, or even how the other species reacted to an alien invasion and if they were able to fight it off. We can only speculate as to why the Leviathans/Reapers made the Rachni do this.
You're still missing the point.
I make no assumptions about why the rachni would invade Citadel space. The reason has no bearing on my argument. They did, however, invade Citadel space and started conquering worlds inhabited by other species.
Oh, the possibilities. To name just a few:
#1) The rachni are a naturally hostile, aggressive, territorial, imperialistic species. Their response to the salarians opening the relay make perfect sense.
#2) The rachni are a peaceful, friendly species who thought their salarian visitors looked yummy and ate them. Curious about the new (Citadel) space on the other end of that freshly opened relay, they sent a small scout team to check it out. The scouts found entire worlds full of yummy salarians - and other species. With such abundant food supplies on offer, those peace-loving rachni started settling those worlds.
#3) The rachni are - whatever they are, and currently under the control of the leviathans/reapers, who have them doing... whatever they're doing. If the leviathans/reapers want them to invade Citadel space - for whatever reason - they aren't doing it. They aren't opening the relay. They aren't looking for other ways to get there, even though they're pretty good at finding things like the Mu Relay. Of course, if the leviathans/reapers want them to go to Citadel space, they could probably tell them exactly how to get there. But now - oh, the salarians opened a relay. So now, the leviathans/reapers suddenly decide to have them enter Citadel space and start conquering their worlds.
Basically, we're asked to believe that whoever was singing the sour yellow note had the rachni go all hostile and aggressive on the other Citadel races just as the salarians opened the relay.