Search Console's coverage report plus a plain site query tells you whether indexing is the problem. If fewer than two thirds of your canonical URLs are indexed, stop here: crawl acceleration cannot force a page the index has already filtered. Fix duplicates, thin pages and orphan URLs first, then re-test in two weeks.
Server logs are the only honest source here. Count hits that carry a real crawler user agent, map them to the URL set you care about, and compare against Search Console's crawl stats. A page returning 200 with fresh content but seeing fewer than a handful of visits a month is the actual candidate for crawl acceleration.
Run an internal link audit and map the path from the home page to your money pages. Anything that needs four or more hops, or that is reachable only from a sitemap, gets crawled less often. Pull a few orphan pages back into navigation before paying for extra visits.
Crawl budget is not free. Pages made of thin boilerplate, duplicated category listings and no unique answer consume visits that should go to commercial pages. Consolidate or noindex those first, otherwise you are buying attention for content you do not want indexed.