While this has some advantages (low pressure, no fission products in the FLiBe), it also some issues.
First, the fuel cycle costs are higher than a LWR. The fuel is dispersed as small encapsulated grains in graphite spheres. Manufacturing the fuel is more expensive, I believe the enrichment needed is higher, and the volume of the spent fuel is considerably larger. All that graphite needs to be disposed of along with the spent fuel.
Second, FLiBe require isotopically separated lithium. Li-6 has a ruinously high thermal neutron absorption cross section so it must be rigorously excluded. It also produces tritium when it absorbs neutrons, which would permeate through the reactor and beyond. But there are no large scale lithium isotope separation plants in operation, and the technology that was used for this in the Cold War (to make Li-6 for H-bombs) has been shut down and cannot be restarted because of mercury pollution (liquid mercury is an inherent part of the process and much escaped down drains at Oak Ridge.)
Kairos has announced operation of a FLiBe purification plant, which sounds promisingly like an isotope separation plant, but it appears it's only a plant for removing other impurities (oxygen, sulfur, iron, etc.) from FLiBe. Isotopically pure Li-7 fluoride would be an input to this plant.
Third, FLiBe is about 11% beryllium. Annual world production of beryllium is just a few hundred tons. There's a limit to how much FLiBe could be made for these reactors (or for fusion reactors, for that matter.)