Adding Surgery and Radiation to the Usual Treatment for HER2-Positive Breast Cancer That Had Already Spread at Diagnosis

Inclusion Criteria: * REGISTRATION STEP 1 - SCREENING \& INITIAL TREATMENT: * Participants must have histologically confirmed de novo metastatic (i.e. American Joint Committee on Cancer \[AJCC\] stage IV, no prior history of breast cancer) HER2+ breast cancer with 1 to 5 distant metastatic lesions (oligometastatic). Metastatic breast cancer must be histologically confirmed through biopsy of a distant metastatic lesion unless it is not feasible or safe to biopsy a metastatic lesion, then there must be unequivocal evidence of metastasis on imaging and laboratory studies. HER2+ status must be confirmed in the breast tumor and distant metastatic site as defined per NCCN guidelines version 4.2025 * NOTE: If HER2 positivity at metastatic site is equivocal due to limitations in HER2 analysis in bone, or it is not feasible or safe to biopsy the metastatic lesion, then HER2 positivity based on breast tumor only is acceptable. If there is no breast lesion, HER2+ must be confirmed in at least one metastatic site * Participants with brain metastases are eligible if they meet all of the following criteria at baseline: * There are 5 or fewer intracranial lesions * Each intracranial lesions is no larger than 2cm in longest dimension * NOTE: The brain is counted as 1 distant site towards the oligometastatic definition. Brain imaging is not required for participants who have no neurological signs or symptoms suggestive of central nervous system (CNS) involvement; however, assessment of neurological symptoms and brain MRI is strongly encouraged to rule out CNS disease * Participants must have no known leptomeningeal disease * Participants must meet one of the following criteria prior to Step 1 registration: * Treatment naïve and planning to initiate standard of care systemic HER2+ targeted treatment OR * Have already initiated standard of care systemic HER2+ targeted treatment and have completed at least one (≥ 1) but no more than three (≤ 3) cycles prior to enrollment, without progression * NOTE: Standard of care scans for baseline disease assessment must have been performed within 28 days prior to initiation of treatment * Participants must not be receiving or planning to receive any investigational systemic therapy during study before disease progression * Participants must not have received whole brain irradiation * NOTE: Participants with up to five brain metastases may have received SRS/SRT either before or after initiation of systemic therapy. If lesions are small and asymptomatic and the participant is receiving CNS penetration they may be managed with observation * Participants must be ≥ 18 years old at the time of registration * Participants must have Zubrod performance status of 0-2 * Participants with known human immunodeficiency virus (HIV)-infection must be on effective anti-retroviral therapy and have undetectable viral load test on the most recent test results obtained within 6 months prior to registration * Participants must be able to safely receive the physician's choice of standard of care systemic HER2+ targeted treatment per the current Food and Drug Administration (FDA)-approved package insert(s), treating physician's discretion, and institutional guidelines * Participants must be candidates for definitive surgical and radiotherapeutic management of locoregional disease opinion per the discretion of the treating physician * Participants must be able to receive ablative dose stereotactic body radiation therapy (SBRT) to all metastatic lesions identified at baseline, in the opinion of the treating physician * Participants must be offered the opportunity to participate in specimen banking * NOTE: As a part of the Oncology Patient Enrollment Network (OPEN) registration process the treating institution's identity is provided in order to ensure that the current (within 365 days) date of institutional review board approval for this study has been entered in the system * Participants must be informed of the investigational nature of this study and must sign and give informed consent in accordance with institutional and federal guidelines For participants with impaired decision-making capabilities, legally authorized representatives may sign and give informed consent on behalf of study participants in accordance with applicable federal, local, and CIRB regulations * REGISTRATION STEP 2 - COHORT ASSIGNMENT AND RANDOMIZATION: * COHORT A: Participants must be registered to Step 2 within 14 days of staging scans performed after completing Step 1 systemic therapy * COHORT A: Participants must have standard of care (SOC) staging scans and breast imaging performed following 12-24 weeks (minimum of 4 cycles) of initial systemic therapy and within 14 days prior to Step 2 registration * COHORT A: Participants must have experienced partial or complete response in the breast (in the opinion of the treating physician) following 12-24 weeks of initial systemic therapy based on staging scans performed within 14 days prior to Step 2 registration * Participants with breast lesion(s) at diagnosis must have a clinical response on breast imaging per the treating physician * Metastatic lesions, as assessed by Response Evaluation Criteria in Solid Tumors (RECIST) and bone response criteria, must have at least stable disease * Brain metastases treated with SRS/SRT must have at least stable disease as per Response Assessment in Neuro-Oncology Brain Metastases (RANO BM) and must not require steroid treatment * Participants with no known breast lesions at diagnosis must have at least partial response in the metastatic sites as assessed by RECIST and bone response criteria * NOTE: The same imaging modality must be used as in Step 1 * COHORT A: Participants must be eligible for definitive surgical and radiotherapeutic management of locoregional disease per the discretion of the treating physician * COHORT A: Participants must not have metastatic lesions that are not amenable to abl