Actually now that I think about it, destroying the Batarian system didn't slow down the Reapers much if at all and conveniently eliminated a huge fighting force of Batarians without the Reapers having to lift a single finger. So that was either a really poor judgment call, or an indoctrinated decision.
You're gravely underestimating the importance of that relay. You're also overestimating the batarians.
Destroying the relay wasn't just to buy a little time. It was also to prevent them from using the relay to get to every part of the galaxy immediately. The relay was what is referred to as an 'alpha relay'. Instead of going into details that can be found on the wiki I'll just simplify it:
Normal relays are limited in the distance that they can send you. You may not realize this just by playing the game but you don't just take one relay jump to get anywhere in the galaxy. You just only see one jump - there are actually multiple jumps required depending on your destination. This is why the reapers take so long in the game to fully take over all the regions. When you highlight a place in the galaxy map you'll notice it paint a blue line from your location to the destination, kinda like connect-the-dots. This is indicating the number of relay jumps you're going to be taking to reach the target location.
"There are two kinds of mass relay, primary and secondary. Primary relays can propel a ship thousands of light years but only link to one other relay, its "partner". Secondary relays can link to any other relay over shorter distances, only a few hundred light years."
The alpha relay that Shepard destroyed was special. It, unlike the normal relays, has far superior range. It would enable you to jump your ship to the citadel and many of the other relays that would otherwise be impossible to get to with one jump. If the reapers were to get this relay they could literally jump right over to the citadel and take over the way they usually do. Remember the method Reapers prefer to use when harvesting: they take over the citadel and shut down the mass relays to systematically harvest the galaxy system-by-system. They isolate us and harvest us... thank the spirits for the protheans modifications to the keepers, huh?
Destroying that relay not only kept the reapers from getting here early but also saved us from the reapers having basically a straight shot into all our systems.
As for eliminating a huge batarian fighting force... 300,000 isn't that significant when you're fighting reapers. Plus over half those were slaves! Yeah, the army of beaten batarian slaves are going to rise up and save us from the reapers. They can't even save themselves, the poor bastards, how are they going to stop the reapers?
Wiki population: Estimated 90,000 (free), 215,000 (other)
Given the batarian's history of slavery I think its a safe bet as to what the 'other' entails. Though the wiki does mention there are military installations, so maybe they'd be able to survive a week. Though I doubt it given how quickly the reapers were able to pretty much harvest their homeworld which has considerably more inhabitants and undoubtedly greater military defense.
Point being: destroying that relay was the only thing that gave us a chance at stopping the reapers and is was absolutely necessary, no matter the politically and moral quandaries that come with.
But it was Shepard's plan to do it even before the Crucible was known to him. He comes to this strategy so quickly I would suspect it was indoctrinated programming-- the inane "Fight as we've never fought before" and "Get reinforcements to Earth!"
While its been pointed out that the conventional warfare thing wasn't Shepard's idea I think its worth mentioning that Shepard's stance throughout the entire trilogy has been to fight the reapers. Inane? Really? Please tell me what alternative earth has at that moment? Should Shepard adopt Saren's doctrine and go "Everyone prove you're worthy and useful to be kept alive! ALL HAIL REAPERS!"
Granted I thought the amount of emphasis put on "save earth" rather than "build crucible" was silly I don't see anything silly about FIGHT OR WE DIE. There is no other alternative. Unless you're Liara, apparently, who suggests you could probably hide for at least their lifetimes before the reaper harvest is complete since it takes so long. Personally, between running, pleading and fighting I have to say fighting seems the most consistent with Shepard. Besides, its not like the whole earth could just up and hide away somewhere. At this point their options seem to be to either obey the reapers and hope for special treatment (as they offer in their indoctrination camps) or fight tooth-and-nail. What else could Shepard have said? Pray?
Also, as others pointed out, Arrival is not canon. Though it probably should be... it would make the trail seem more important. If you didn't do Arrival I think you're being held because you worked with Cerberus. Like, seriously? REALLY? I can understand the relay incident but... lol wow.