Cisplatin and Gemcitabine Hydrochloride With or Without Berzosertib in Treating Patients With Metastatic Urothelial Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed metastatic urothelial carcinoma; urothelial cancer derived from the bladder, ureter or upper tract is permitted * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 20 mm (\>= 2 cm) with conventional techniques or as \>= 10 mm (\>= 1 cm) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have access to archival tumor tissue for proposed correlative studies; these may be derived from transurethral resection of bladder tumors (TURBT), cystectomy, or biopsy; if archival tissue is not available for proposed correlatives, patients may be enrolled at the discretion of the study principal investigator (PI) (SKP) * No prior cytotoxic chemotherapy for metastatic disease; prior immunotherapy is permitted * At least 12 months have elapsed since platinum-based peri-operative treatment * Karnofsky \>= 70% (Eastern Cooperative Oncology Group \[ECOG\] performance status 0-1) * Life expectancy of greater than 3 months * Leukocytes \>= 3,000/mcL * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100,000/mcL * Total bilirubin within institutional upper limit of normal * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 2.5 x institutional upper limit of normal * Creatinine clearance \>= 50 mL/min by either measured (using the Cockcroft-Gault, Modification of Diet in Renal Disease \[MDRD\] or Chronic Kidney Disease Epidemiology \[CKD-EPI\] formula) or calculated clearance (i.e. glomerular filtration rate \[GFR\]) * The effects of M6620 (VX-970) on the developing human fetus are unknown; for this reason and because DNA-damage response (DDR) inhibitors as well as other therapeutic agents used in this trial may have teratogenic potential, women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; 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; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 6 months after completion of M6620 (VX-970) administration * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Radiotherapy within 4 weeks of protocol therapy * Patients who are receiving any other investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to M6620 (VX970), cisplatin, or gemcitabine * M6620 (VX-970) is primarily metabolized by CYP3A4; therefore, concomitant administration with strong inhibitors or inducers of CYP3A4 should be avoided; because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated medical reference for a list of drugs to avoid or minimize use of; Patient Drug Information Handout and Wallet Card should be provided to patients; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements * Pregnant women are excluded from this study because M6620 (VX-970) as a DNA-damage response (DDR) inhibitor may have the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with M6620 (VX-970), breastfeeding should be discontinued if the mother is treated with M6620 (VX-970); these potential risks may also apply to other agents used in this study * Patients with \>= grade 2 neuropathy