There are some all-composite business jets, like the newest Bombardier Learjet. The design choice is about trade-offs in manufacturing: supply chain, number to be built, available tools and workers, legal certification (old methods are considered proven so require far less testing data), existing techniques, cost/percentage of build flaws, market price of the end product, needed performance and lifespan.
Then there is a question of available qualified and certified after market mechanics for repair, inspection and maintenance. Downtime is required to repair minor damage like bird strikes and examine internal damage from loose carts or cargo that is a visible dent on metal but only looks like a scuff on composite but could be internal delamination. (Boeing had to consider and develop solutions for all of these external demands. Some solutions only made sense for large carriers with major maintenance divisions.)
For example, if a significant mistake is made on a lamination layer, the whole composite part is pretty much landfill garbage, while an aluminum panel or rib can be recycled and replaced as fairly low cost and effort.
On the other side, business jets like Lear generally have far different flight profiles over their life compared to commercial jets, operate at different cabin pressure, are more concerned with altitude and speed, may consider small airports a constraint, and pennies per seat mile is a much lower consideration. Cost of downtime for repair or inspection is a very different priority as well.
Then there are the aspects of fire, repainting (paint stripping specification), effect of fuel any that may leak into the structure, and of course weather certifications for lightning and icing conditions. For de-icing and anti-icing, aluminum can be easily warmed from the inside while plastic is a poor conductor and may need another method. For lightning protection the 787 has a metal cloth for the outer layer and electrical bonding between all the parts, and this adds weight and thickness; as a percentage, a widebody jet is less affected by the extra material than a narrow body (the circumference doubles but the cross section area is quadrupled). Not all private aircraft are required to have full lightning protection. If they will not be flying near such conditions, maybe just the fuel tanks will have static and lightning protection. Also, useful payload is far less of a concern for business jets than for regional carriers who often sell extra baggage capacity to freight or parcel carriers like UPS/FedEx.