Why doesn't Jon Snow just create a new wight to show to Cersei?

Question

In Game of Thrones season 7 episode 6, "Beyond the Wall", Jon Snow and several other important characters feel the need to

go ranging beyond he wall to capture a wight.

This is (obviously, and as it turns out) very dangerous. However, in Season 1, Othor's body is risen from the dead south of the Wall, and attacks Jeor Mormont, so obviously the White Walkers' influence works that far.

So instead of going north of the Wall for their mission, couldn't they just have put some already chained corpse next to the Wall and waited for it to be raised as a wight? Even if the Wall prevents that, going 10 meters north of it with a corpse would have been much easier, and

would probably have prevented Viserion's death.

Why didn't they do that instead?

(I understand there might be some thics involved about raising someone from the dead on purpose. That would probably cause problems to Jon, but not necessarily to Tyrion, Daenerys, The Hound, etc., so they could at least have argued over this...)

Answer 1

First off Othor was already a wight when he came through the Wall, there is a continuity error with how the bodies are shown but they are clearly already turned.

Jon Snow: Just these two, My Lord. Been dead awhile, I'd say.
Jeor Mormont: Mmm.
Samwell Tarly: The smell.
Othell Yarwyck: What smell?
Samwell Tarly: There is none. If they'd be dead for a long time, wouldn't there be rot?
Jon Snow: We should burn them.
Othell Yarwyck: Snow's not wrong, My Lord. Fire will do for them. The Wildling way.
Jeor Mormont: I want Maester Aemon to examine them first. You may be a coward, Tarly, but you're not stupid. Get them inside.
_{Game of Thrones, Season 1 Episode 8, "The Pointy End"}

However, their eyes aren't actually blue yet, as they should be, but I think this is an error on the shows part. Actually when Othor attacks Mormont his eyes have gone blue now.

This is also backed up by the books in that Othor's eyes are blue when he comes through the Wall.

"Othor," announced Ser Jaremy Rykker, "beyond a doubt. And this one was Jafer Flowers." He turned the corpse over with his foot, and the dead white face stared up at the overcast sky with blue, blue eyes. "They were Ben Stark's men, both of them."
_{A Game of Thrones, Jon VII}

Your question is also based off of a secondary false premise in that a dead person does not turn into a wight automagically it is a manual process as we saw at Hardhome with the Night King showing off in front of the group on the boats.

So why wouldn't this have worked? Well we don't know the White Walkers max limit to their power yet so it's likely they wouldn't have even been able to reach the dead. Secondly I think it's safe to assume that the White Walkers wouldn't have even known the person had been killed for them to be able to turn them. Lastly, the White Walkers appear to be a separate intelligent species so I doubt they'd turn someone to a wight when it's clear what the end goal would be.

Answer 2

Because only White Walkers can reanimate bodies. Bringing a corpse beyond the wall will not make it a wight. As for Othor, he already had blue eyes when he was found, so he was already resurected when the NW found him.