If the only thing known to be wrong is a date, the sort of thing kids can easily get confused about, I don’t think that should lead to negating bigger things.
Also, from what you say here:
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I was always sure that my brother had his transplant on my mum's birthday due to what i remember from stuff she would say
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Your mother herself is the source of the mistake. If she made big dramatic statements about how the transplant was the universe’s birthday gift to her or some such, most people would think it happened on her birthday.