the centrifugal force and the weight are applied in different points.
all the ~100km of tether below geostationary orbit will have positive net weight by definition. All this weight must be held by tensile strength, whether it comes from a fleet of rockets or centrifugal force is irrelevant.
Again the hard thing is not making it float, geostationary satellites do this already, the hard thing is building a 100km long satellite.
EDIT: said another way consider a chain being pulled apart by two tractors. All forces on the chain cancel each other yet some chains will break and some will not.
Forces are first applied, then propagated, then (vectorially) summed. If the material cannot handle the forces applied it will break.
If you happen to have seen those "Hydraulic press against X" videos you can see how it is not enough to have forces cancel each other in different points.
And to be clear about how this is relevant: for (at least) the first 50km or so the centrifugal force is negligible, so even if
And in order for a space tether to work at all, that weight must be negated by centrifugal force or the entire thing will collapse to the ground.