How necessary is? The use of a knee brace after a total knee arthroplasty?

Usually not needed. Knee replacement rely on the integrity of the medial and lateral (inside and outside) ligaments. All total knee replacements substitute for the anterior cruciate ligament, and many for the posterior cruciate as well. If a brace is used after surgery, it is generally because the medial ligament is thought to be weak, or is injured in the course of the operation.
Not generally. The knee replacement is stable at the time that it is performed. We use your own ligaments and tendons and do not generally harm them during surgery. There are cases where the ligaments (such as the mcl) are stretched during surgery, were moved and reconstructed, or were weak to begin with. In these cases, the knee brace is critical after surgery.