The idea of a flat-Earth is a parody in itself. At times, I doubt that anybody seriously believes in it. There are plenty of plausible reasons why somebody might publicly claim to believe in such nonsense: To build a strong opposition to authority, feel special, see people's breathless reactions when you tell them about your persuasion, and even have a second-tier explanation. To put it shortly, it is fun.

In the case of actual flat-Earth believers, some of the reasons above could play a role, needless to say, a subconscious one.
Nevertheless, common sense is not powered by better knowledge but by disinterest. How many people could actually argue for a spherical Earth? Keep in mind how often academically educated people get elementary physics and mathematics wrong. Realistically, at least half the population of an enlightened country like the United Kingdom can be expected to have no clue how to argue for a globe. They simply never question it and therefore never need a proof. Naturally, many of my own assumptions about how the world works are like that.
A mathematician (as long as he or she isn't a logican or set theoretizist) or engineer would never be in the awkward situation of asking themselves, »Wait, is 1 + 1 actually 2? Is this just a matter of definition, or something? Is there a proof for it?«

If allowed, I did like to return to the point about the second-tier explanation. What do I mean by this?
A first-tier explanation would, in this context, be a straightforward explanation: This event happened because of another event that came before it, and so on. A second-tiere would be »in the context of this ideology« or »given this presumption, we can explain«. It is a kind of hypothetical thinking, a »what if«, not a »just-so« explanation.
From a logical point of view, it may not even count as an explanation at all. However, an observer of the human psyche will notice some strange instances in which this kind of explanation actually holds, even if the subject knows that the framework is made up or even wrong. There is a story Yudkowsky (a writer of Harry-Potter-Fanfiction) tells about a woman who initially lost her faith but regained it by forcing herself to live according to her sect's behavioural code.

One might very well think something like, »Ah, the subconscious cannot distinguish between first- and second-tier explanations, then. That, or something like it, is what you aimed to tell us, right?«
I would answer: »You can understand it in that specific way. Or you could say that 'explaining something' is a special behavior learned through conditioning, just like other behaviors. If there is positive reinforcement, it can be strengthened, just like any other behavior«.