As a Scout, I wouldn't want to take credit from something I didn't actually earn or deserve. I wouldn't want to encourage a Scout to do that either. Becoming a member of the OA merely due to a clerical error made during a call out ceremony is meaningless and should not be what that honor represents... but sending a Scout back to his campsite humiliated, disenfranchised and in tears due to an adult's mistake is hardly what the Scouting program should stand for either.
In such a situation, I would explain to the Scout what happened (off to the side, to avoid any public embarrassment), and I'd let him advise us on what we should do. He may opt to recuse himself until he is properly elected in... but maybe not. If he decides to stay, I would encourage him to go through all the steps to get into the Order legitimately, even if the steps he's taking are somewhat out of order ("okay, you were called out, so you can go through the ordeal tonight; but to be fair to all the other Scouts that wear the sash, afterwards we should make sure you get voted in by your peers and let's make sure you've completed all the required nights camping, reached First Class, etc."). That way when he wears the sash and pocket flap and attends OA events he can say he did truly deserve the honor listed on his Scouting résumé and he does deserve to be there (even if his pathway there was unorthodox or backwards).
I had a Scout several years ago that accidently got signed-off on Citizenship in the Nation merit badge at summer camp when he didn't actually complete all the requirements (due to a paper-work mix-up with the councilor at the camp, he got credit for writing a letter to a congressman when he never actually did it). I didn't withhold the badge (nor the advancement to the rank of Life that earning the badge entitled him to). But I did encourage him to actually go do that requirement he had skipped... which he did. Now he wears his Eagle badge with pride and knows he did everything the badge requires (even if he technically did them out of order). If he hadn't written the letter, I wouldn't have withheld further rank from him; but he'd know that he took a shortcut and I think his Eagle would be less meaningful to him.
So, in this case, I wouldn't withhold membership or participating in the ordeal due to an adult mistake during the callout, but I'd encourage the Scout to make the honor meaningful by going back to dot the 'i's and cross the 't's.