Randomized Phase II Trial of Lutetium Lu 177 Dotatate Versus Everolimus in Somatostatin Receptor Positive Bronchial Neuroendocrine Tumors

Inclusion Criteria: * PRE-REGISTRATION: Pathologic Documentation: Well- or moderately-differentiated neuroendocrine tumor(s) of bronchial origin (i.e. carcinoid) as assessed by local pathology * The pathology report must state ONE of the following: * Well- or moderately-differentiated neuroendocrine tumor, * Low- or intermediate-grade neuroendocrine tumor, or * Carcinoid tumor (including typical or atypical carcinoid tumors) * PRE-REGISTRATION: Documentation of histology from a primary or metastatic site is allowed * PRE-REGISTRATION: Functional (evidence of peptide hormones and/or bioactive substances associated with a clinical hormone syndrome such as carcinoid syndrome or Cushing's syndrome) or nonfunctional tumors are allowed * PRE-REGISTRATION: Patients with poorly-differentiated or high-grade neuroendocrine carcinoma (i.e. large cell neuroendocrine carcinoma of lung, small cell lung cancer) or mixed tumors (i.e. adenocarcinoid tumor) are not eligible * PRE-REGISTRATION: Recurrent or locally-advanced/unresectable or metastatic disease * PRE-REGISTRATION: Neuroendocrine tumor of bronchial (i.e. lung) primary site * PRE-REGISTRATION: Lesions must have shown radiological evidence of disease progression in the 12 months prior to pre-registration * Tumor must have shown somatostatin receptor (SSTR) positivity on 68Ga-DOTATATE PET or other SSTR-PET scan in the 12 months prior to pre-registration; however, documentation of SSTR positivity in the 6 months prior to pre-registration is preferred. SSTR positivity is defined as uptake greater than background liver in all measurable lesions * PRE-REGISTRATION: Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 by computer tomography (CT) scan or magnetic imaging (MRI). Any lesions which have undergone percutaneous therapies or radiotherapy should not be considered measurable unless the lesion has clearly progressed since the procedure * PRE-REGISTRATION: Lesions must be accurately measured in at least one dimension (longest diameter to be recorded) as \>= 1 cm with CT or MRI (or \>= 1.5 cm short axis for lymph nodes). Non-measurable disease includes disease smaller than these dimensions or lesions considered truly non-measurable including: leptomeningeal disease, bone metastases, ascites, pleural or pericardial effusion, lymphangitic involvement of skin or lung * REGISTRATION: Confirmation of SSTR positivity by Alliance Imaging Core Lab (ICL) at Imaging and Radiation Oncology Core (IROC) Ohio central radiographic review * REGISTRATION: Patients with treatment-naive or previously-treated disease are allowed. Patients with previously-treated disease must have demonstrated radiographic disease progression on the prior therapy * REGISTRATION: No prior treatment with peptide receptor radionuclide therapy (PRRT) (e.g. lutetium Lu 177 dotatate) * REGISTRATION: No prior treatment with mammalian target of rapamycin (mTOR) inhibitors (e.g. deforolimus, everolimus, sirolimus, temsirolimus, etc.) * REGISTRATION: Prior treatment with hepatic artery embolization (including bland embolization, chemoembolization, and selective radioembolization) or ablative therapies (i.e. cryoablation, radiofrequency ablation, etc.) is allowed if measurable disease remains outside of the treated area or if there is documented disease progression in a treated site. Prior liver-directed (including ablative) treatment must be completed at least 28 days prior to registration * REGISTRATION: Prior treatment with 90-Yttrium radioembolization must be completed at least 6-weeks prior to registration * REGISTRATION: Radiation therapy to the lung and/or mediastinum must be completed at least 14 days prior to registration for stereotactic ablative and at least 28 days prior to registration for conventional fractionation * REGISTRATION: Prior treatment with systemic anticancer therapy must be completed at least 28 days prior to registration (except for somatostatin analogs in patients with functional tumors). Continuation of treatment with somatostatin analogs while on protocol therapy is allowed provided that the patient: * Has functional tumors (evidence of peptide hormones and/or bioactive substances associated with a clinical hormone syndrome such as carcinoid syndrome or Cushing's syndrome), and * Has previously demonstrated radiographic disease progression while on somatostatin analog therapy * REGISTRATION: Patients must have completed any major surgery at least 28 days prior to registration. Complete wound healing from major surgery should occur prior to registration * REGISTRATION: Patients should have improvement of any toxic effects of prior therapy (except alopecia, fatigue, and other non-reversible toxic effects such as neuropathy from cisplatin) to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 5.0, grade 1 or less * REGISTRATION: Not pregnant and not nursing * Therefore, for women of childbearing potential only, a negative pregnancy test done =\< 28 days prior to registration is required * REGISTRATION: Age \>= 18 years * REGISTRATION: Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * REGISTRATION: Hemoglobin \>= 8.0 g/dL * REGISTRATION: Platelet count \>= 75,000/mm\^3 * REGISTRATION: Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * REGISTRATION: Creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated creatinine clearance \>= 40 mL/min * Calculated by the Cockcroft-Gault equation * REGISTRATION: Total bilirubin =\< 2.0 x ULN * In patients with Gilbert's syndrome, if total bilirubin is \> 2.0 x ULN, then direct bilirubin must be =\< 2.0 x ULN * REGISTRATION: Albumin \>= 2.8 g/dL * REGISTRATION: Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 3.0 x ULN * REGISTRATION: No known central nervous system metastases unless treated and clinically stable for at least 14 days prior to registration. Patients on steroid support must be clinically stable on weaning doses of steroids * REGISTRATION: No other currently active malignancy that requires therapy or is expected to require therapy during the study (excluding non-melanoma skin cancers or in situ carcinomas, such as breast or cervical) * REGISTRATION: No known active hepatitis B (defined as hepatitis B surface antigen \[HbsAg\] reactive) or known active hepatitis C virus (defined as hepatitis C virus \[HCV\] ribonucleic acid \[RNA\] viral load detected). The exception is for patients with known active hepatitis B virus (defined as HbsAg reactive) infection, where the HBV viral load must be undetectable on suppressive therapy for patient to be eligible * REGISTRATION: Patients with known human immunodeficiency virus (HIV) infections on effective anti-retroviral therapy with undetectable viral load within 6 months of registration are eligible for this trial * REGISTRATION: No known active or uncontrolled infections requiring ongoing antifungals or antibiotics in the 3 days prior to registration * REGISTRATION: No receipt of live attenuated vaccines in the 7 days prior to registration * REGISTRATION: No known decompensated liver cirrhosis * REGISTRATION: No known prior drug-induced pneumonitis that was symptomatic or required treatment * REGISTRATION: No known medical condition causing an inability to swallow and no known impairment of gastrointestinal function that may significantly alter the absorption of an oral agent * REGISTRATION: No known hypersensitivity to everolimus or other rapamycin analogs (e.g. sirolimus, temsirolimus, etc.) * REGISTRATION: Concurrent somatostatin analog use while on protocol therapy is allowed provided that the patient: 1) has a functional tumor (evidence of peptide hormones and/or bioactive substances associated with a clinical hormone syndrome such as carcinoid syndrome or Cushing's syndrome), 2) has previously demonstrated radiographic disease progression while on somatostatin analog therapy * REGISTRATION: Chronic concomitant treatment with P-gp and strong CYP3A4 inhibitors and/or inducers is not allowed on the everolimus treatment arm of this study. Given that the study is randomized, all patients on P-gp and strong CYP3A4 inhibitors and/or inducers must discontinue the drug(s) 7 days prior to registration * RE-REGISTRATION: Confirmation of disease progression by RECIST v1.1 by real-time Alliance ICL at IROC Ohio central radiographic review * RE-REGISTRATION: Not pregnant and not nursing * Women of childbearing potential only, a negative pregnancy test done =\< 28 days prior to re-registration is required * RE-REGISTRATION: ECOG performance status 0-2 * RE-REGISTRATION: Hemoglobin \>= 8.0 g/dL * RE-REGISTRATION: Platelet count \>= 75,000/mm\^3 * RE-REGISTRATION: Absolute neutrophil count (ANC) \>= 1,500/mm\^3 * RE-REGISTRATION: Creatinine =\< 1.5 x upper limit of normal (ULN) OR calculated creatinine clearance \>= 40 mL/min * Calculated by the Cockcroft-Gault equation * RE-REGISTRATION: Total bilirubin =\< 2.0 x ULN * In patients with Gilbert's syndrome, if total bilirubin is \> 2.0 x ULN, then direct bilirubin must be =\< 2.0 x ULN * RE-REGISTRATION: Albumin \>= 2.8 g/dL * RE-REGISTRATION: AST/ALT =\< 3.0 x ULN