Doing your own research also means being open to the possibility that your hypothesis is incorrect.
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We know what the OP is attacking, and generally agree. Effectively saying that commoners can't do their own research isn't the way to get that done.
You need more practical advice than just "you can't do it". Something like the CRAAP test mentioned here is closer to the right approach.
Thinking about it more, something simpler makes more sense.
If you're into talking to a flat earther, I'd recommend getting a little into flat earth stuff. You'll understand it better if you're looking at the same sources they are (valid or not). And, of course, vice versa. And you can then choose which one makes more sense, maybe even switching if the opposite side seems to make more sense than your own.