Changing the default search engine to something other than Bing and configuring search on new tabs to use the address bar causes the selected default engine to be used on new tabs, both in the address bar and in the search box.
However, when running in InPrivate mode, new tabs always use Bing search in the search box ("InPrivate search with Microsoft Bing"). This happens regardless of the configured default engine, and even if Bing is removed from the list of search engines. Searching from the address bar, instead, does use the configured default search engine, as expected.
Sighted with v81.0.416.68 under Windows 10 and Windows 7.