It was quite the journey. First, I accepted that it was possible that the authorities could be lying to us for some reason. Next, I decided that a FE was possible. Finally, after much internal struggle, I came to the shocking conclusion that a FE was probable.
My process:
1. In 2007 when not much information was available on the topic my first goal was to find out as much about FE as I could. I purchased a copy of Earth Not a Globe and read it from cover to cover with captivation. It was very engaging. These days you would probably be better off reading our Wiki for more modern information, but ENAG still has unsurpassed commentary.
2. My next goal was to decide if FE was possible, independent of any argument someone might have in favor of it (ie. Rowbotham might have gotten some stuff wrong, but this would be irrelevant to actual the shape of the Earth, so you would have to decide for yourself). This was done for me through debate and discussion. The RE arguments were fairly weak when assessed critically and couldn't preclude the possibility of a FE. Seek further evidence, or ask a question against the narrative, and it all falls apart.
3. My final goal was to decide if FE was probable. To do this I went through all of the topics and compared the RE explanation to the FE possibility. I decided that there were a lot of coincidental things and insufficient RE explanations, such that the whiteboard of FE possibility won out in the end. Each of the subjects discussed in the Wiki is a letter. The letters spell out a sentence. The sentence is: THE EARTH IS FLAT
It shouldn't be possible to think that a FE is possible, yet it is. Physical science should be absolutely contradictory to an FE, yet it's not. There should be no cherries to pick, no physicists to cite, and nothing to give sway to these positions, yet there is plenty of evidence in their favor.
Whenever I post a wiki link the weak arguments against it are 'cherry picking!' and 'bias!', and not actual contradictory evidence which completely devastates it. Simply put, this complete lack of an argument means that we won the argument. It's all pretty boring now. The main thing the RE have going for them in support of their arguments is their personal incredulity, which only erodes as the years pass and additional data is collected.