Scroll saws are at their best cutting wood 3/4" and thinner. They can cut thicker wood, but not easily. I think most can cut up to 1 1/2", but that is a pita. They also have a limited cutting capacity, with the depth of the table's throat limiting the area that can be cut. If a cut is farther from the edge of the board than the depth of the throat, it cannot be cut with that scroll saw. Also, scroll saws are not the tool of choice for cutting straight lines. That is what a table saw or track saw is for.
If I were building fishing rod holders, I would choose the saw based on the thickness of the wood, which would probably be 3/4", so a scroll saw would be a good choice for any curved areas. You could cut the straight parts with the scroll saw and then plane or sand them to the straightness required. The main reason I would choose the scroll saw for this over a band saw is that the edges cut on the scroll saw require minimal sanding, mainly to remove the fuzzies on the corners of the cut. If you are going to rout the edges, then even that is not necessary. With a band saw, all the edges would need sanding to remove the saw marks and smooth the curves, which can take a lot of time. This could also be cute with a jig saw, but the edges again needing sanding.
For more on choosing a scroll saw, you can look for some of the videos by Steve Good on You tube. He and many others have discussed this topic in great detail.
Tom Wilson