This phase II/III compares the standard therapy (chemotherapy plus cetuximab) versus adding bevacizumab to standard chemotherapy, versus combination of just bevacizumab and atezolizumab in treating patients with head and neck cancer that has spread to other places in the body (metastatic or advanced stage) or has come back after prior treatment (recurrent). Immunotherapy with monoclonal antibodies, such as atezolizumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Bevacizumab is in a class of medications called antiangiogenic agents. It works by stopping the formation of blood vessels that bring oxygen and nutrients to tumor. This may slow the growth and spread of tumor. Cetuximab is in a class of medications called monoclonal antibodies. It binds to a protein called EGFR, which is found on some types of cancer cells. This may help keep cancer cells from growing. Cisplatin and carboplatin are in a class of chemotherapy medications known as platinum-containing compounds. They work by killing, stopping, or slowing the growth of cancer cells. Docetaxel is in a class of chemotherapy medications called taxanes. It stops cancer cells from growing and dividing and may kill them. The addition of bevacizumab to standard chemotherapy or combination therapy with bevacizumab and atezolizumab may be better than standard chemotherapy plus cetuximab in treating patients with recurrent/metastatic head and neck cancers.