Is it possible to shrink wrap terrain to a placed static mesh?

Define shrinkwrap.

Also, in short you just cant.
The landscape uses a maximum of 1vert per meter.
There is no guarantee a mesh vertex will fall anywhere near that landscape vertex.

You can look into heightfiled mesh as a possible alternative. 9 years down the line ofc.

Back then, you just used meshes.

Even now, you should still just use meshes.