I would like to suggest "motivational bias" as an alternative name; it is a more accurate description than "motivated cognition", which is much too general.

I don't see spending too much time investigating things you don't want to be true as too bad a problem, a bit wasteful, but that's it. "Motivated stopping" is more likely to lead you astray, you need to remember to keep questioning things you agree with. In fact, this is a common bias that is often used by con artists, for example, and a good rule is a generalization of the common self-defense against con artists contained in the phrase, "If it looks to good to be true, it probably is." If you are immediately tempted to accept or agree with an argument, take a second look.