Actually, I once visited a website that didn't close its tags where that practice hindered my use of it. That page used backticks instead of curly quotes, so I wanted to use a bookmarklet to fix that. The bookmarklet cycles through every DOM text node and replaces the text. When I tried to run it on that page, nothing happened – probably because there were no text-only nodes, because the page was treated as one big tag. So browsers don't handle missing closing tags completely correctly, and therefore one should include closing tags.
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Actually, I once visited a website that didn't close its tags where that practice hindered my use of it. That page used backticks instead of curly quotes, so I wanted to use a bookmarklet to fix that. The bookmarklet cycles through every DOM text node and replaces the text. When I tried to run it on that page, nothing happened – probably because there were no text-only nodes, because the page was treated as one big tag. So browsers don't handle missing closing tags completely correctly, and therefore one should include closing tags.