The original is making a point that your rewrite neglects.
[...] to show that the accuser did not simply lack sufficient evidence
to obtain a conviction, but actually[...]
This clause is fully exculpatory. It says that we understand that simply lacking sufficient evidence carries zero cause for recompense. Whereas -
[...] to show that the accuser not only lacked sufficient evidence to obtain a
conviction, but actually[...]
Suggests the two situations are just of differing degree, hence both might be cause for recompense. This is simply not true.
Simply is being used and an emphasizer in the original, and it has a different effect than only. Not only X, but also Y is a standard bit of conjunctive grammar that isn't intensive. But you have proposed *Not only X, but actually Y. This doesn't really work actually. We use not simply, but actually to sell the idea that two things are distinctly different, not merely of differing degree.
Not simply displaying language, but actually realizing it in order to achieve some kind of purpose or other.
That is, if somebody is going wrong in some way, we should not simply
point out the fact that they are 'going wrong', but actually help make
it clear what they need to do [...]
Not simply, but actually asks the reader to make a clear distinction between the two things being compared.