I've been told the lack of certain HTML tags in Web Designer websites such as < H1 > or < li > hinders search engines.

This is not true at all. Search engines index the text and do not take into account any specific HTML tags when indexing your website. The text, not the tags or the level of heading, is what matters and is indexed. Many experts have even suggested that keyword tags are ignored by the likes of Google.  What affects your ranking in the search engines is having good quality, relevant, on topic text, and having a good website structure (i.e. page linking). But more than this, what matters is having good quality (preferably highly ranked) websites linking to your website (this is the Google page rank system).
To quote Google: PageRank relies on the uniquely democratic nature of the web by using its vast link structure as an indicator of an individual page's value. In essence, Google interprets a link from page A to page B as a vote, by page A, for page B. But, Google looks at more than the sheer volume of votes, or links a page receives; it also analyzes the page that casts the vote. Votes cast by pages that are themselves "important" weigh more heavily and help to make other pages "important".

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