The simple answer is no – search engine submission isn’t necessary. The majority of search engines nowadays (most notably Google) crawl and index pages by following links. Using that logic, a single inbound link from any already-indexed page will identify your page to the engine. Subsequently, if that page links to other pages within your site, they will also be indexed… and so on.
This is one of the reasons backlinking is still such an important facet of SEO. But the more backlinks your website has is not always better! Backlinking in 2021 is much different than it was in 2015. Today, backlinking is a quality over quantity game. The idea being that if your content is worth something to the end user, there is a natural tendency for others to share links to said content.
In any case, submitting your site to the search engines certainly does not hurt. Plus, it only entails about 5 minutes of added work. You generally don’t need to submit your sitemap more than once. It doesn’t get you indexed any quicker if you submit more than once.
Also note, submitting your sitemap to search engines is entirely FREE. Many SEO companies insist that you need to pay to be indexed. This is simply not true.