Well, first, the idea that someone will conclude they are being irrational is mainly a rationalist pipe-dream. People might conclude they are wrong about something, and sometimes they might decide they are being overly emotional, but they almost never think they are being irrational about it. People always have their reasons, and reasons are always couched in some worldview that establishes what's good and right and worthwhile. It's far, far harder than you might think to convince someone their good, right, worthwhile worldview is irrational.
Further, we should all recognize (though few do) that rationality is a power mode, not all that dissimilar to physical force, social influence, legal/moral authority, etc. If we are intent on using rationality to force people into compliance — the way a physically strong person might intimidate another into submission, or a person with influence might crowd-shame another to obey - then we are starting off on the wrong foot. Any power can be abused, including the power of rationality.
When we have intractable philosophical questions, the intractability is rooted in emotions. Stubborn philosophical questions arise because someone decides to intellectually challenge some deeply held belief. It's interesting to note that every notion listed in the question — solipsism, cogito ergo sum, the 'hard problem' of consciousness, philosophical zombies, religious belief, a priori arguments for or against God — digs into the oh-so-deeply-held belief that we humans are something more than sum of our biological parts (a window into the question asker's soul, perhaps, if souls exist). People feel self-aware, independent, conscious, free; other people seem to experience the same things; we all ascribe that hard-to-avoid feeling to some source (usually some metaphysic). It's worth looking into those feelings and ascriptions to see if we can make anything of it. I'm not certain we need to solve such questions in any absolute sense. They create good positions for self-reflection, contemplation, and self-development, and we don't have to worry about them further unless they are resolved by advances in the physical sciences at some future point.
But such questions are troublesome to the extent they represent seats of power. To the above example, much of what we take as good and right in the world is built on those feelings and ascriptions, and many social institutions and positions owe their existence to them. Here is where we get into serious tangles. Religious leaders condemn secular law because it runs against their religious authority. Atheists attack the notion of the soul to dispute religious authority, then quickly backtrack to rehabilitate free will, without which we cannot have secular authority. Young idealists denounce political leadership as amoral and mercenary, while old partisans decry the lack of patriotism and honor in rebellious youth.
This shift into power-tangles creates a different type of intractable question; people engage these because they are trying to establish authority more than they are trying to establish truth (often, mind you, because they mistakenly believe that authority is what establishes truth), and people worried about authority can be very stubborn indeed. But this kind of intractable question can resolve through dialogical philosophy (such as the Socratic method), where we actively avoid asserting power through reason. Dialogical philosophy draws out the reasoning of the other and puts it through its paces (in the manner of Wittgenstein's 'therapy') in order to get down to its human (not philosophical) roots. When we humanize these questions (and thus humanize the person we are facing) the power-tangle dissolves, and we're left with the philosophical question itself.
Bit wordy, this; apologies…
