In this sentence, what does "not wanting" mean?

This piece is written in old fashioned English. It's almost certainly 19th century if not 18th. At that time the word wanting was often used to mean that there was a shortage of something.

We do still use wanting in this way but much less frequently and usually in the phrase found wanting which means that someone or something had not performed well.

In this case, however, wanting is negated (not wanting) so the writer is saying that there is no shortage of circumstances which go a long way to prove his or her point. This addresses the first part of your question, there are ENOUGH circumstances to prove the point.

The second part is simpler to explain. The phrase 'brushing aside the conclusions of science' is surrounded by commas which means that it is parenthetical (it is additional to the sentence like this part in brackets). If you ignore the parenthetical part the second part of the sentence becomes "...that instinct felt that it had rightly come upon the truth."

This means that felt relates to instinct and means that instinct is given the human characteristic of feeling that it had discovered the truth.

As I said this is very old fashioned English and is, to a modern reader, an over-complicated and obscure piece of writing. If you have to read it for some purpose then you need to analyse it but don't be tempted to try to write like this; modern English does not, fortunately, work that way. Having said that I suspect that it wasn't terribly clear when it was written either. Please edit your question to give the source of the quoted passage.