So from reading this Wikipedia summary it looks like the situation was actually more complicated than either of us realized (although definitely closer to your summary):

FitzRoy had found a need for expert advice on geology during the first voyage, and had resolved that if on a similar expedition, he would "endeavour to carry out a person qualified to examine the land; while the officers, and myself, would attend to hydrography... he asked his friend and superior, Captain Francis Beaufort, to seek a gentleman naturalist as a self-financing passenger who would give him company during the voyage. A sequence of inquiries led to Charles Darwin, a young gentleman on his way to becoming a rural clergyman, joining the voyage

So yes, he was self-financed. But the primary issue is that he was given support by the crew and the entire existence of an exploration ship (which if anything seems pretty similar to the space program you've criticized). Would you call someone who does work without pay now using data from NASA an amateur? If so, then the term "amateur" isn't very relevant to capturing the most important detail- where the resources for their work comes from.