Why does it matter that search engines use other search engines by proxy? Using DDG for Bing, or Startpage.com/Ixquick for Google, or Searx for either of them,

Why does it matter that search engines use other search engines by proxy? Using DDG for Bing, or Startpage.com/Ixquick for Google, or Searx for either of them, should not provide the original search engine with your data. Bing and Google should just see actually anonymous search queries coming in from DDG / Startpage / whatever you're using, instead of seeing the search requests coming from your browser session. At least this is my understanding of it - please correct me if I understand this wrong. But if this is the right way to understand it, I don't get how using DDG would then be equivalent to using a Microsoft product directly.