Someone deleting an answer with a +3 score is potentially not a good thing, if the answer actually added value and they're just after the badge.
I don't see much point in the badge myself. I don't see that the tiny amount of potentially good behaviour that could possibly be encouraged by it outweighs the potential loss.
The main for arguments given so far are:
sometimes one of them is really not contributing anything
It encourages people to delete incorrect answers, even if they have a score of 3.
An answer with +3 is not likely an incorrect one and is likely to be adding some value. Even if that value is simply a different wording that a handful of people understand more than another answer's wording.
I'm not saying that we should have many variants of the same thing, of course, but arguably, it added value to at least 3 people, even if they just felt it added value to the site!
Is this not what user votes are for and are to signify?
If this is failing, then it's not the badges fault, nor can the badge rectify this, most certainly so being a one-hit-wonder!
If however the +3 answer is deemed as not valuable, then it should be downvoted, and then the user should edit it.
Either way, a "one off" badge shouldn't be the tactical approach to maintaining this issue. Downvotes, comments, and the user removing it due to downvotes are the key players here.
Clean up - does the badge cut it?
Sometimes old questions have over time gathered a few answers, often the answers are pretty much the same (certainly in terms of +value). In this scenario, perhaps deleting an answer with +3 to "clean up" the question is worthwhile.
However, we're not talking about how valuable site clean up is, or users removing dupe answers to help make a question less cluttered (etc).
This question matter was in relation to the specific badge being pointless, and if the above quoted responses are the only arguments for the badge, then the badge is pointless as it's only awarded once - that is, each user only does a "single clean up" once.
There must be better ways to approach this.
If you could win the badge multiple times, perhaps it would be of use for getting users to clean up.
Otherwise, while it may be argued to help, (and yes I'm speculating here) I would bet that there are a lot of people deleting a +3 answer only for the badge and where their answer was actually valuable.
Compared to the small number of users doing it because they thought their answer was not adding value and had a surprise when the badge was given?
It's surely no contest?
You have to also consider that a lot of the clutter and poor answers are simply made by people who do not care.
Users who won't clean up, regardless of a badge or not, and certainly wont be removing their precious +3 voted answers any time soon.
Proof of harm
This is tag specific, but in PHP tag, certainly these days, +3 on a basic question often signifies the answer one should be reading.
It shouldn't be removed.
Of course the badge being there could add some value even by an occasional user cleaning up to get the badge, but again, how many delete an answer which had value?
I still think the badge has more potential to do harm than it could good, especially as the good it can do is so insignificant.
All that said, I certainly don't think it warranted you (Esq) to be so, well, passionate, that you edited this question enough times to make it community wiki, nor to the extent that you argued with people as you did. e.g.:
such a waste, 50 people voted on this stupid question while they could
have actually helped someone. posts that get even 20 votes in a couple
of hours are really rare on stackoverflow.