Should I submit my website to search engines for indexing?
Search engines used to offer submission forms for the websites but they're no longer available.
Instead, Google proposes to send your sitemap.xml through the intermediary Google Search Console. In the meantime, the Bing form was removed in 2018.
You should rather try to get links pointing to your website. These new links will attract the search engines robots more efficiently.
Moreover, if your internal pages are not indexed, then it would probably be beneficial for your website that you focus on the optimization of your internal links by strengthening your internal linking.