Testing Whether Treating Breast Cancer Metastases With Surgery or High-Dose Radiation Improves Survival

Inclusion Criteria: * A patient cannot be considered eligible for this study unless all of the following conditions are met. * Pathologically confirmed metastatic breast cancer * Known estrogen, progesterone, and HER2 status of either primary tumor or metastasis; * Note: estrogen, progesterone and HER2 status of metastasis preferred for stratification * Number of allowable metastases: * =\< 4 metastases seen on standard imaging within 60 days prior to registration when all metastatic disease is located within the following sites: * Peripheral lung * Osseous (bone) * Spine * Central lung * Abdominal-pelvic metastases (lymph node/adrenal gland) * Liver * Mediastinal/cervical lymph node * All known disease amenable to metastasis-directed therapy with either SBRT or resection * Note: Symptomatic bone metastasis are allowed if ablative therapy can be delivered * Note: Sites for possible surgical excision include lung, liver, adrenal gland, bone, small intestine, large intestine, ovary, and amenable nodal disease sites * Note: Surgical stabilization is allowed for a metastasis if it is followed by conventionally fractionated external beam radiotherapy * Maximum diameter of individual metastasis in any dimension =\< 5 cm * There are no restrictions on distance between the metastases * Patients must be registered within 365 days of the initial metastatic breast cancer diagnosis; first-line standard systemic therapy (chemotherapy, anti-endocrine therapy, anti-HER2, or other standard targeted therapy) for metastatic breast cancer must be given or planned to be given; if given before study entry, it cannot have exceeded a duration of 12 months at the time of registration (Note: sequencing of ablative therapy \[surgery or SBRT\] relative to systemic therapy, for patients randomized to Arm 2, is at the discretion of the treating physician) * The primary tumor site must be controlled prior to registration * For those who present with synchronous primary and oligometastatic disease, primary must be controlled prior to registration * The definition of control is definitive surgery by excision or mastectomy (+/- radiotherapy) per institution preference For those who present with local recurrence and oligometastatic disease, local recurrence must be controlled prior to registration * The definition of control is definitive surgery by excision or mastectomy (+/- radiotherapy) per institution preference * Appropriate stage for study entry based on the following diagnostic workup: * History/physical examination within 60 days prior to registration * Clinical grade computed tomography (CT) scans of the chest, abdomen, and pelvis with radionuclide bone scan OR whole body positron emission tomography (PET)/CT within 60 days prior to study registration * Zubrod performance status =\< 2 within 60 days prior to registration * Blood cell count (CBC)/differential obtained within 60 days prior to registration on study * Absolute neutrophil count (ANC) \>= 500 cells/mm\^3 * Platelets \>= 50,000 cells/mm\^3 * Hemoglobin \>= 8.0 g/dl (note: the use of transfusion or other intervention to achieve hemoglobin \[Hgb\] \>= 8.0 g/dl is acceptable) * For females of child-bearing potential, negative serum or urine pregnancy test within 14 days prior to study registration * The patient or a legally authorized representative must provide study-specific informed consent prior to study entry Exclusion Criteria: * Patients with any of the following conditions are NOT eligible for this study. * Pathologic evidence of active primary disease or local/regional breast tumor recurrence at the time of registration; * Co-existing or prior invasive malignancy (except non-melanomatous skin cancer), unless disease free for a minimum of 3 years; previous RT dose, date, fraction size, must be reported * Metastases with indistinct borders making targeting not feasible * Note: A potential issue with bone metastases is that they often are not discrete; since many patients on this protocol will have bone metastases, this will be an important issue; theoretically, Houndsfield units might provide an appropriate measure; however, a sclerotic lesion against dense cortical bone will not have a sharp demarcation based on Houndsfield units (HU); therefore, we acknowledge that such determinations will pose a challenge and thus the physician's judgment will be required * Prior palliative radiation treatment for metastatic disease to be treated on the protocol (including radiopharmaceuticals) * Metastases located within 3 cm of the previously irradiated structures: * Spinal cord previously irradiated to \> 40 Gy (delivered in =\< 3 Gy/fraction) * Brachial plexus previously irradiated to \> 50 Gy (delivered in =\< 3 Gy/fraction) * Small intestine, large intestine, or stomach previously irradiated to \> 45 Gy (delivered in =\< 3 Gy/fraction) * Brainstem previously irradiated to \> 50 Gy (delivered in =\< 3 Gy/fraction) * Whole lung previously irradiated with prior percent volume receiving greater than or equal to 20 Gy (V20Gy)\> 30% (delivered in =\< 3 Gy/fraction) * Primary tumor irradiated with SBRT * Metastasis irradiated with SBRT * Brain metastases * Exudative, bloody, or cytological proven malignant effusions * Severe, active co-morbidity defined as follows: * Unstable angina and/or congestive heart failure requiring hospitalization within the last 6 months * Transmural myocardial infarction within the last 6 months * Acute bacterial or fungal infection requiring intravenous antibiotics at the time of registration * Chronic obstructive pulmonary disease exacerbation or other respiratory illness requiring hospitalization or precluding study therapy at the time of registration * Pregnancy; lactating females must cease expression of milk prior to signing consent to be eligible * Human immunodeficiency virus (HIV) positive with cluster of differentiation (CD)4 count \< 200 cells/microliter; note that patients who are HIV positive are eligible, provided they are under treatment with highly active antiretroviral therapy (HAART) and have a cluster of differentiation 4 (CD4) count \>= 200 cells/microliter within 30 days prior to registration; note also that HIV testing is not required for eligibility for this protocol