The specific fallacy is cherry picking evidence to support a conclusion.
This is one of the most common fallacies committed by people with actual intention to be rational and is based on the invalid belief that the way one should arrive at a view is through supporting verifications. There was a dominant philosophic movement based on the Verification Principle, Logical Positivism, which held that only verified beliefs were valid, and that science was based on verifications.
Karl Popper pointed out that verifications are FAR TOO EASY to arrive at for all sorts of invalid beliefs, as your examples show. The way science should be done, and by extension, basically all thinking, is by looking for possible REFUTATIONS of a view or hypotheses. Hence, in science, the key to showing a new claim is valid, is to do experiments which could DISPROVE it. If the claim passes those experiments, then we have reason to trust it is valid.
For non-science reasoning, when one isn't doing experiments, the challenge is instead to consider all of the possible sources of evidence that could bear on a judgment, not just one. Therefore, the justification for politician, person, or religion to be "good" should NOT be -- "X is good because of fact Y", but "X is good because: looking at the full spectrum of relevant issues, X is good at Y, Z, Q, R, and S, and this is sufficient to compensate for X's failings on A, B, C and D; because of the relative importance of the latter part of the alphabet over the earlier (or whatever standard one thinks is appropriate -- this too will need comparative evaluation); AND X is better than other X's because all X's are imperfect; and this X has more strengths and fewer failings than is common (again will need extensive justification by looking at refuting cases)."
Popper's focus on challenging one's own views rather than confirming them, has some venerable support in philosophic history. This was basically the method of Socrates. Socrates was not an experimentalist, but his method of philosophy was to try to identify the hidden and unexamined assumptions in one's views, and try to challenge them, by looking for inconvenient facts that show them to be invalid over generalizations. In my Philosophy Café, I say this is the central philosophic mindset -- to identify and question the walls of the mental boxes we think within. We may choose to keep those walls, at least most of the time. But examining and challenging them, to know they are there, and are not as certain as we presumed, is key to understanding where our own thinking may be going awry.