The sentence is currently written as a negation :
I do not think [this] but [I think that]
Straight substitution could sound confused and appear less right and less logical - as if you had two conflicting thoughts (even though the substance of the thoughts is identical). If you wanted to use "while", the sentence could be written :
Some people think that either nature or nurture is the major influence
on a person, while I think that both have powerful effects.
So the simultaneous event is different people having different thoughts.
Or you could do without the conjunction :
I do not think that either nature or nurture is the major influence on
a person. I think that both have powerful effects.
Having a separate sentence for each thought avoids the simultaneous "this is wrong and that is right" which, even if this and that are different, is expressing both agreement and disagreement in the same sentence - though "this is wrong but that is right" doesn't have the same problem as it is already a negation.