This phase III trial compares the effect of adding surgery to a standard of care immunotherapy-based drug combination versus a standard of care immunotherapy-based drug combination alone in treating patients with kidney cancer that has spread to other places in the body (metastatic). Immunotherapy with monoclonal antibodies, such as nivolumab, ipilimumab, pembrolizumab, and avelumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Axitinib may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Surgery to remove the kidney, called a nephrectomy, is also considered standard of care; however, doctors who treat kidney cancer do not agree on its benefits. It is not yet known if the addition of surgery to an immunotherapy-based drug combination works better than an immunotherapy-based drug combination alone in treating patients with kidney cancer.