Object | Weld is essentially a ‘special case’ of Face | Bridge - it was never intended to be used (or capable of) welding individual verts - because of the way wings works.
Verts left selected after the weld op act as an indication of the extent of the faces operated on during its use.
One of its main intended purposes is to allow the user to (easily) bridge / join 2 separate objects that share a common face in intimate (or nearly so) proximity.
Under these conditions, Face | Bridge can also be used - but selecting the 2 faces to be bridged (if they’re in contact with each other) isn’t particularly easy (unless you use selection growth techniques, or store the faces prior to the op) - hence the arrival of Object | Weld onto the scene.
No need to select faces to be bridged, just select whole objects and press ‘go’
More info here if interested