would the satellites placed on a Lagrange orbit be more stable?

Probably not, it would still be perturbed over time by those forces (not Lunar tides, though, so perhaps a little more stable).

Orbital dynamics are really a lot more jumbly and chaotic than intuition might suggest, over long timespans. The current arrangement of the solar system only seems stable because it's had billions of years for all the really unstable stuff to get shaken out. Unoccupied regions of space are usually unoccupied because there's some mechanism that disturbs anything that tries to hang around there long-term.

Finding defunct satellites would probably not be too difficult for a civilization at our level. They're small, but they're always nearby so there's plenty of opportunity to catch them in surveys. You probably wouldn't want to use the James Webb, though, for a couple of reasons:

• James Webb is in the Earth-Sun L2 point, so to observe objects in the vicinity of Earth it'd have to point toward the Sun. That would wreck its ability to keep itself cool, which is vital for an infrared telescope, and blind the optics if you weren't being extra careful when pointing it.
• James Webb is designed to look at distant objects with high resolution, so its field of view is very small. It's like looking through a straw, you're going to have a hard time catching things that you don't already know where to aim for.

Something like the Simonyi Survey Telescope is better suited to this. It's designed to have a wide field of view so that you can regularly blanket the sky with photographs, then compare the photographs from different times to see if anything's moved relative to the distant stars. Those moving things are likely to be stuff like asteroids and satellites.

Ironically, geostationary satellites would be the hardest type to detect like this since they don't move relative to the distant stars. But since they'll get pushed out of geostationary orbit over time by the perturbations I mentioned, that future civilization will still be able to find them if they're still somewhere nearby.