How people select truths is a fascinating and complicated subject. While philosophy gives us tools to analyze beliefs, particularly through the study of logic, the philosophy of mind has only since the advent of cognitive science as a discipline really started to unpack the relationship between feelings and reason. There are two claims that are becoming increasingly clear with modern reasearch:
- People are not Homo economicus and are irrational in many ways including cognitive biases.
- Beliefs are an extension of one's emotions and are susceptible to cognitive dissonance, rationalization, and originate in the gut as per the somatic marker hypothesis.
Non-cognitivism as a meta-ethical theory (IEP) on its face makes a claim that complex ethical arguments essentially reduce to simplified emotional claims. That great sermon given by your pastor that ran 45 minutes long? It's simply "Yay for the Golden Rule!". MLK's I Have a Dream reduces to "Boo for racism!" While the Geach-Frege Problem and other objections have some technical weight, it's hard to escape the conclusion that philosophical arguments are rooted in first principles that often have emotional significance. Children are experts at reasoning from conclusion to premises, for instance to get what they want.
It's an empirical question of why many people reject non-cognitivism (despite the strengths of its position and consistency with an empirical exploration of how people reason practically), however, it's a rather a truism that by definition rationalization as a psychological activity is a defense mechanism to protect us from our feelings. Therefore, a person engaged in rationalization must behave in a way as to be oblivious to the act of rationalization itself. Unlike philosophical bullshit, a person who rationalizes does have a concern for the consistency of truth-conditions, because the emotional import of the truth and falsity of statements is what is at stake in the reasoning process.
Think about how people rationalize to avoid the emotional impact of their inevitable death. If one wants to avoid Camus's project of confronting death head on, and hides in comforting myths of a religion, then the best way of doing that is to engage in a lengthy and distracting process of sifting through claims, reason, and truth conditions to mitigate the impact of the truth. But if you reduce their position to "Jesus yay, because death boo", you bring back to attention the premise of inevitable death. This is such a common tactic of death with uncomfortable and inconvenient truth, that the Buddha made a religion out of pointing out that it is not Right Thinking.
That people routinely use language to obfuscate and avoid truth is part of the language-game in the Wittgensteinian sense. People lie, palter, use false implicature, obfuscate, use rhetorical tactics, advertise, bluff, conceal, and distract routine to affect claims of truth. This is part of how language works since it has illocutionary, locutionary, and perlocutionary effects. Non-cognitivism strips away some of the illusion and ambiguity of language and drills down to people's authentic feelings, and attempts to make plain people's agendas. People who abuse language to avoid truth and honesty, and that's a fair number of people, as well as people who persist with a cognitive distortion about how the world really works find the reduction of complex ideas to simple feelings do not want or cannot deal with such vulnerability.
So, while there are some technical objections to non-cognitivism, the great objection, in my estimation, seems to be that people have beliefs about reason which overestimate its power and effect. People believe they have more control then reason actually affords. People resent having all of their complex logical arguments reduced to simple value propositions. People refuse concede to the extent of their own irrationality. Thus, the objections are more of a psychological nature than a philosophical one. Thus, most people simply stick to "Reduce my thinking to my feelings? Boo!"