To make this explicit here, we're dealing with two different kinds of velocities. The equations of relativity always deal with the velocity of an object as measured by a specific observer, or more precisely, in a specific inertial reference frame.
The velocity in the first part of SpaceTiger's example is a different kind of velocity. We didn't measure that velocity "directly"; we inferred it from the difference in the velocities of the two objects, as measured by us. People often call this a "separation velocity" or "closing velocity" depending on whether which direction the objects are moving.
Classically, the separation or closing velocity of two objects as measured by a third observer equals the velocity of either object as measured by the other one (ignoring signs). In relativity this is not true. To get the velocity of either object as measured by the other one, from the velocities of the two objects as measured by the third observer, you have to use the relativistic velocity-addition formula.
There's nothing fundamentally wrong with the separation velocity. Sometimes it's a perfectly sensible quantity to use. It's just not the kind of velocity that the relativistic transformation equations apply to.