Answer for the question Why are planes using rivets & not welded construction? describes why rivets are preferred for joining aluminium alloy components. But how are carbon-fibre-reinforced plastic components in new aircraft joined?
Ideally, fiber composites are bonded and connected by a scarf joint. The scarf ratio is determined by the ratio of fiber tensile strength and the shear strength of the resin. If your fiber takes 3000 N/mm², the resin transmits 20 N/mm² and the fiber content is 60%, the ratio is 1:90, so a 1 mm thick composite layer would need to be joined over a length of 90 mm. In reality, such a ratio is impractical to achieve, so lap joints are used when accepted by the certification authorities. This is the case for for composite gliders, where you will find both scarf and lap joints.
In the early days of powered composite aircraft, fiber composites were seen with the experience of aluminium in mind, and treated with the same methods. Therefore, the Lear Fan and its contemporaries had to use rivet joints throughout the structure. The idea was that by riveting the airframe could be more easily disassembled for inspection and repairs. Riveting carbon fiber composites is quite problematic: You need to use expensive titanium fasteners because aluminium rivets would quickly corrode when in electrical contact to the electrochemically much nobler carbon fibers. Drilling and countersinking holes cuts the very fibers which you try to connect, so you have to double up on wall thickness what is destroyed by the riveting. But the FAA still insisted on rivets - even stringers on panels were riveted, when bonding would had been the most sensible solution.
Aircraft engineers have an expression for composites that are treated like metal: Black Aluminum. The LearFan was pure Black Aluminum. Many of the engineers in this project ended up with Beech when they produced the Starship, so many of the same techniques were transferred as well. At least, the Starship had a fuselage shell which was laid up on a mandrel and then cut just aft of the cockpit section to get the mandrel out. The splicing of the two fuselage sections was then done by a tapered lap joint; if you want, a practical way of making a not-quite scarf joint.
Today we have a new generation of engineers who slowly accept that the nature of composite structures requires their own methods of treatment. But you will still find riveted joints even in modern composites structures. Old habits are hard to break.
What about 'Iron Bridge'?
joints were similar to what a carpenter would use. So using traditional method for a new material, iron. They seem to be doing the same with carbon fiber.
Time lapse videos of 787 show riveting for the main fuselage.