I think it is impossible because surface tension will prevent solder from filling unplated vias especially small ones. As FR4 material repells solder.

Of course you can. But assembling with wire is easier. The trouble I suspect is heating both sides of the board simultaneously such that you can get the solder to flow on both sides, then overcoming the surface tension of the solder and getting it to go through the hole.

To use solder is much faster than a wire. Imagine you have 100 or 200 vias which you have to fill. I think either way is OK as long as the speed does not concern you.

Thanks for marvelous posting.