The question is vague, so it can be several different things. Generally, dismissing an argument based on who is supporting it is called ad hominem, "attack on an argument made by attacking the character, motive, or other attribute of the person making the argument, rather than attacking the argument directly". It might also be what is more specifically called converse appeal to authority "that must be false, because that's what [infamous expert] says".
However, I am not sure what is meant by "otherwise valid", and "support it for the wrong reason". If their reason for supporting it is part of their argument, and it is wrong, then the argument is at least unsound (has false premises), or perhaps even invalid, if the wrongness affects the reasoning. If proponents who present "partial and weaker (and easier to refute) representation" of the argument are deliberately picked out this would be the selection form of the straw man, which Talisse and Aikin call "weak man", and in extreme forms Drum calls "nut picking", i.e. cherry picking nutty supporters.
EDIT: Based on the added example, this seems closest to the ad ignorantiam, appeal to ignorance, ("ignorance" refers to lack of contrary evidence), concluding that X because presented arguments for not X are all flawed. Of course, it is not necessary that they are flawed specifically due to circular reasoning. The fallacy illicitly shifts the burden of proof to the opponents by asserting one's own position as the default. It is a case of false dilemma, because it excludes the possibility that presented evidence is insufficient to draw a conclusion either way. The proverbial rebuttal is the oft-quoted "the absence of evidence is not the evidence of absence", or in this case, the absence of good arguments to the contrary is not an argument for their absence. In the example another false dilemma appears in the presumption that the Earth is either flat or round.