It isn't a double negative when the negatives are in different clauses.
[I didn't say] [he wasn't coming].
There is nothing wrong with that.
We can't turn that last one into "I said he was coming". That is not the intended meaning.
Is the intended meaning is ' he wasn't coming' but the news wasn't coming from my mouth. Is that right?
That is one interpretation. In that case a speaker would emphasize the word "I".
Another possibility is that he is correcting the assumption of another person.
A: Jim called. He is having car trouble.
B: Oh, it's too bad that he won't be coming.
A: I didn't say that he wasn't coming. If he can get the car started, he might show up.
Putting those two concepts together, it is correct to say:
[That is not to say] [there is no problem] and [no need for a solution].
Now I think this means 'There is no problem and there is no need for a solution'. But the fact was not said by 'that' .
The problem with this is that this sentence and the paragraphs I cited are not logical because if there had had no problem and no need for a solution people would not have sought to control a few unwanted species. Is that right?
No. And this is tough. In the first part of the paragraph, the writer is complaining about the use of toxins to control insects. He states his reasons for the objection. That might lead the reader to assume that he doesn't think there is a problem and he doesn't think anything should be done. But he doesn't believe that. He recognizes that there is a problem and that something needs to be done. So he transitions to that (and probably a safer strategy) by writing "that is not to say that there is no problem...." By doing that he clarifies his approach to the issue.
1. The current approach is wrong.
2. The fact that the current approach is wrong doesn't mean that an approach isn't needed.
3. Here is my suggestion.