That installing Google Analytics 1 or verifying Search Console 1 helps indexing or rankings is unconfirmed, with no support in Google's documentation. Pages "can be discovered by Google" without submitting a sitemap, and sites rank without either tool.
Use both for what they measure. Search Console shows Google's view (queries, clicks, impressions, average position, indexing, Core Web Vitals, security issues and manual actions) and emails you about new issues. Google Analytics 4 shows what visitors do; Universal Analytics stopped processing data on July 1, 2023. Google Tools tours both.