English Translation
One can’t learn to do anything without making mistakes.
What connotation does this sentence have over 「人はミスを犯して、何事でもできる。」
In other words, what additional meaning does 「ようになるものだ」add?
ようになるものだ per se is grammatically correct and it’s a quite common phrase. ようになる = Someone becomes more X. / Someone turs into X.
e.g. 私の妹は自転車が乗れるようになった。= My little sister became to be able to ride a bicycle.
～ものだ is conjugated when the speaker wants to describe a pattern, a principle, or a theory from an objective perspective with a definitive/scientific/philosophical tone. So, you cannot insert ～ものだ into 私の妹は自転車が乗れるようになった because some people cannot acquire an ability to ride a bicycle. But 何事もできるようになる + ものだ is a correct and natural collocation because the speaker generalizes the lesson. It’s an observed pattern throughout our human history.
The original Japanese sentence doesn’t sound natural, though… Particularly, 何事でもできる is wrong. It should be 何事もできる. I would propose:
失敗は成功の母（しっぱいはせいこうのはは）or 失敗は成功のもと
This is actually a well-known Japanese proverb (see the definition). The literal translation is “making mistakes is the mother of success”.
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