I'm confused on the intended purpose of the french drain in your scenario. Is the intent to help drain the outside of a foundation wall of a house? Or to help dry out a soggy part of a yard? Or divert water away from the top or bottom areas of a retaining wall? The exact reason this line-of-thought would be wrong depends on these specifics.
None-the-less, it is likely wrong.
French drains work based on the permeability of the surface and immediate sub-surface. In a typical french drain, a trench is created that is typically deep and somewhat narrow. This provides an area of easier flow for water to drain from surrounding areas to the drainage pipe at the bottom.
Your walkway example would not have the drain pipe to eliminate the water from the area and allow drainage to flow away from the area.
Further, french drains are typically covered by a very thin layer of soil (2-4") to allow the surface water a quick entry into the pipe below. Because in most instances the french drain is located in a low area, this area naturally holds water and needs a quick escape to prevent waterlogged terrain.
Of course, if not that much water needs to be removed, french drains are possible without the drain pipe. These hold only as much water as the volume available of the trench minus the rock. In permeable subsoils, these function well for quickly holding smaller amount of water and then allowing them to slowly seep into the subsoil. If your subsoil is less permeable, you'll want a pipe to remove the water.
In a typical sidewalk, the area of gravel is relatively shallow and wide. If your sidewalk had 3" of gravel, they would need to be 4' wide to have the same volume as the narrow, deeper trench, but this gravel will be compacted for even less water holding capacity. Further, this would only be adequate for surface runoff. If you have any concerns of subsoil saturation, the deeper trench would be needed to help the deeper soils to drain as well.
If this is for a wall or foundation, the drains must be at the lowest level of those features and drain the water well away from them. Even with a walk out basement, your foundation is likely to be 1-3' below the depth of the flooring of the lowest house level. The sidewalk would not be deep enough to drain the to the extent needed.
There is a possibility of combining the two features, of course. Dig a deeper trench as would be typical to whatever depth was needed for the footer of the wall, and construct the drain as typical for your need. However, instead of filling the trench to grade, stop 4-8" below grade to the level of the sidewalk base material and construct the sidewalk as normal, however, with the base gravel 3-6" wider than the walk. After the concrete sets, replace the sod around the sidewalk, building up the gravel thickness if needed to keep sod depth minimal with good draining soils (especially if you poured thicker than average walks) This will allow water to quickly reach the drain on either side of the sidewalk.
So - it is possible if you have very limited requirements, but for almost all cases, you'll need to create both the drain and the walk, but perhaps they can have the same footprint. Hope this helps!
Edit: If you are not concerned about draining the foundation but instead just the surface flow, then I think this would work best with a ridged pipe included under the base for the concrete (or pavers) and a quick migration path from the surface to the base and flushing from beneath. Depending on the climate, I'd be worried about freeze damage to the walk way at the end of run without providing extra runout to ensure the water can travel well off the walkway areas prior to freezing. Finally, usual french drains will hold and transport more per ft3 due to the compacted nature of the fill - 1" gravel provides a lot of space between the aggregate that sand and other paver base would simply not provide. Of course, the paver base would, to a lesser extent, move water better than the soil alone. It entirely depends on how much water you want/need to move.