Why don't gas flames radiate much heat directly, but metal objects heated by them do?
The luminous flame itself is hot; the site where the chemical process of combustion takes place. But the product of combustion, typically CO₂, H₂O, and perhaps some CO gas carry off a great quantity of the heat energy created in the reaction.
The main purpose of the lava rocks, grill etc. is to capture a good part of that heat flux so it doesn't just blow by the food into the atmosphere. Another reason is to provide a more uniform distribution over the cooking field.
The reason the lava rocks 'feel' hotter than the bare flame is that they have the capacity to store a lot of that heat. The flame and hot product gases transfer heat over time and that heat integrates over time to raise the temperature of the rocks.
A hot solid body emits more heat energy (principally in the form of infrared radiation) than the flame heating it, first, because the solid has more atoms than the flame. More precisely, it has more atoms visible (i.e. from which electromagnetic radiation may pass unobstructed) to the place where the radiated heat is detected; and the individual atoms, or more exactly their electronic shells, are the source of emission of the radiation. Moreover the atoms in the solid are held in a matrix, and so the heat energy forces their electrons to jiggle about and become effective sources of radiation. In a hot gas, much of the heat is in the form of translational kinetic energy of small molecules as they whizz unimpeded (for most of the time) through space; and this high-speed travel, of itself, doesn't result in the emission of radiation. There are further differences in the radiative effectiveness of solids versus gases that result from their particular chemical composition---and, in the case of the solid, from the physical conformation (e.g. rough or smooth) of the surface, from which most of the radiation is emitted.
The 'lava rocks' have two advantages over a simple gas flame. First, they can catch drippings and give that nice smoky aroma. Any fat that falls through a gas flame just sits on the burner (which is cooled by incoming gas, the flame is slightly above it).
Second, they aren't transparent like the gas from the flame, they're dark. That means they radiate more heat than the flame. Hot gasses from the flame move through the rocks and transfer heat by convection, but the meats and veggies are surrounded by a thin layer of their own steam, which lessens the heat transfer. Rocks, of course, don't steam. They get hotter than the food will, and give off lots of infrared that isn't blocked by steam.
The creation of radiation is also why gas lanterns use a mantle rather than a raw flame. The mantle material glows brighter than the flame.