Testing the Addition of Radium Therapy (Radium-223 Dichloride) to the Usual Chemotherapy Treatment (Paclitaxel) for Advanced Breast Cancer That Has Spread to the Bones

Inclusion Criteria:

* Women or men with metastatic breast cancer with two or more bone metastases identified by technetium Tc-99m (99mTc) bone scintigraphy and/or CT, at least one of these bone lesions must not have been treated with prior radiation therapy
* A diagnosis of breast cancer must have been histologically or cytologically confirmed at any time point
* Patients with non-bone metastases (in addition to bone metastases) are permitted if:

  * Five or less visceral metastasis (=\< 4 cm in size) and asymptomatic (not including lymph nodes)
  * Enlarged lymph nodes =\< 4 cm
* Patients with HER2 negative disease (HER2 negativity by immunohistochemistry \[IHC\] or fluorescent in situ hybridization \[FISH\] ratio according to the American Society of Clinical Oncology-College of American Pathologists guideline criteria) (Hammond et al., 2010; Wolff et al., 2013). Hormone-receptor positive (estrogen receptor \[ER\]-positive and/or progesterone receptor \[PR\]-positive) as well as triple-negative (ER-negative, PR-negative and no overexpression of HER2) breast cancer may be enrolled. Hormone receptor status will be determined at the local institution. ER and PR negativity will be defined as \< 1% tumor staining by IHC
* Patient must be eligible to receive therapy with paclitaxel for the treatment of their breast cancer. Patients with hormone-receptor positive disease should have progressed on at least one prior line of hormone therapy and a CDK4/6 inhibitor in the metastatic setting to be eligible (except if patient had a contraindication or intolerable toxicity with the use of these agents). Previous radiation and chemotherapy for the treatment of metastatic breast cancer is allowed
* Age \>= 18 years

  * Because no dosing or AE data are currently available on the use of radium-223 dichloride in combination with paclitaxel in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2 (Karnofsky \>= 60%)
* Absolute neutrophil count (ANC) \>= 1,500/mcL
* Platelets \>= 100,000/mcL
* Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) (with the exception of \< 3 mg/dL for patients with Gilbert's disease)
* Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional ULN (=\< 5 ULN for patients with liver metastasis)
* Creatinine =\< 1.5 x institutional ULN OR glomerular filtration rate (GFR) \>= 40 mL/min/1.73 m\^2
* Hemoglobin \> 10 g/dL
* Human immunodeficiency virus (HIV)-infected patients on effective antiretroviral therapy with undetectable viral load within 6 months are eligible for this trial
* For patients with evidence of chronic hepatitis B virus (HBV) infection, HBV viral load must be undetectable on suppressive therapy if indicated
* Patients with a history of hepatitis C virus (HCV) infection must have been treated and cured. For patients with HCV infection who are currently on treatment, they are eligible if they have an undetectable HCV viral load
* Patients with asymptomatic, treated brain metastases are permitted if there is no evidence of progression for at least 4 weeks after central nervous system (CNS)-directed treatment, as ascertained by clinical examination or brain imaging (magnetic resonance imaging \[MRI\] or CT scan) during the screening period
* Patients with a prior or concurrent malignancy whose natural history or treatment does not have the potential to interfere with the safety or efficacy assessment of the investigational regimen are eligible for this trial. History of other active malignancy requiring treatment within the last 3 years or bone marrow dysplasia such as myelodysplastic syndrome (MDS) is not allowed
* Patients with known history or current symptoms of cardiac disease, or history of treatment with cardiotoxic agents, should have a clinical risk assessment of cardiac function using the New York Heart Association functional classification. To be eligible for this study, patients should be class 2B or better
* Concomitant use of bisphosphonates or denosumab is required (except if medical contraindication such as hypocalcemia or concern for osteonecrosis of the jaw). If not already on bone modifying agents, patient must initiate such therapy within one month before start of study treatment
* The effects of radium-223 dichloride on the developing human fetus are unknown. For this reason and because alpha particle-emitting radiopharmaceutical agents as well as other therapeutic agents used in this trial are known to be teratogenic, women of child-bearing potential must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry, for the duration of study participation, and for at least 6 months after the last dose. Should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately. Pre-menopausal subjects as well as subjects with ovarian radiation or concomitant treatment with an luteinizing hormone-releasing hormone (LH-RH) agonist/antagonist must have a negative pregnancy test and agree to use an adequate method of contraception as recommended by their treating physicians. Subjects of child-bearing potential who are sexually active and their male partners must agree to utilize, during the treatment period and for 6 months after last dose of radium-223 dichloride, 2 reliable and acceptable methods of contraception used simultaneously: a) barrier method such as a) condoms (male or female) with spermicidal agent or b) diaphragm or cervical cap with spermicide, combined with a highly effective non-hormonal birth control method such as an intra-uterine device. Men treated or enrolled on this protocol must also agree to use adequate contraception and not donate sperm prior to the study, for the duration of study participation, and 6 months after completion of radium-223 dichloride
* Ability to understand and the willingness to sign a written informed consent document. Participants with impaired decision-making capacity (IDMC) who have a legally-authorized representative (LAR) and/or family member available will also be eligible
* No prior paclitaxel in metastatic setting within 2 years prior to radium-223 dichloride start. No prior paclitaxel in adjuvant or neoadjuvant setting within 6 months prior to radium-223 dichloride start

Exclusion Criteria:

* Patients with peripheral neuropathy \> grade 1
* Patients who have not recovered from AEs due to prior anti-cancer therapy (i.e., have residual toxicities \> grade 1) with the exception of alopecia
* Patients who have had chemotherapy or immunotherapy with checkpoint inhibitor within 4 weeks prior to treatment. Patient who receives radiation therapy or hormone therapy within 2 weeks prior to treatment are excluded. For patients on trial therapy prior to study enrollment, washout period of 6 times the half-life of previously administered investigational agents prior to starting radium-223 dichloride is required
* Prior therapy with radionuclides (e.g., strontium, samarium, rhenium, radium)
* Patients who are receiving any other investigational agents. Vaccination for severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is allowed as well as any therapy as required for the treatment of active coronavirus disease 2019 (COVID-19) infection
* History of allergic reactions attributed to compounds of similar chemical or biologic composition to radium-223 dichloride or other agents used in study
* Patients with psychiatric illness/social situations that would limit compliance with study requirements
* Pregnant women are excluded from this study because radium-223 dichloride is an alpha particle-emitting radiopharmaceutical agent with the potential for teratogenic or abortifacient effects. Because there is an unknown but potential risk for AEs in nursing infants secondary to treatment of the mother with radium-223 dichloride, breastfeeding should be discontinued if the mother is treated with radium-223 dichloride. These potential risks may also apply to other agents used in this study
* Imminent/established spinal cord compression, pathological fracture in weight bearing bones or bone lesion with soft tissue component unless treated as appropriate with radiation and/or surgery before starting on trial
* Prior hemibody external radiotherapy
* Patients must not have an active infection requiring systemic treatment
* Patients must not use immunosuppressive medication =\< 7 days of registration, EXCEPT for the following:

  * Intranasal, inhaled, topical steroids, or local steroid injection (e.g., intra-articular injection)
  * Systemic corticosteroids at physiologic doses =\< 10 mg/day of prednisone or equivalent
  * Steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication) is allowed
* Patients with Crohn's disease or ulcerative colitis
* Patients with a marked baseline prolongation of QT/corrected QT interval (QTc) interval (e.g., repeated demonstration of a QTc interval \> 480 milliseconds \[ms\]) (CTCAE grade 1) using Fridericia's QT correction formula
* Patients with a history of additional risk factors for Torsades de Pointes (TdP) (e.g. heart failure, hypokalemia, family history of long QT syndrome)
* The use of concomitant medications that prolong the QT/QTc interval
* Life expectancy \< 6 months