Ignoring exceptions or w.e. I think this one can just be thought of as common sense. A lone carbon is not going to want to give up a proton and deal with a negative charge. The instability is going to lead it to quickly reprotonate as a strong base and thus weak acid. Don't overthink it.
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With all due respect, I don't think any of this is common sense. Your rationale is based on something whether it be assumptions, the fact that you haven't seen it happen often, or a combination of something along with what you know about acids/bases.
It sounds like you're using the strength of the conjugate base for your reasoning, though.
So going with that, comparing atoms in the same period, base strength is opposite electronegativity. Comparing atoms in the same group, base strength is opposite size.
CH4, NH3, and H2O are all in the same period so that makes CH4 the strongest base.
Between PH3 and H2S, PH3 would be the stronger base. Since NH3 is above PH3 on the periodic table, NH3 would be the stronger base since N is smaller in size. But we've already determined that CH4 was stronger than NH3 so the strongest base between all choices would be CH4.
Even though you arrive at the correct answer following these rules, you wouldn't be able to correctly rank all of them since PH3 is actually a stronger base than H2O. So that leaves me back at my previous post...