I think I know now what is happening. . I am actually searching directly using the address bar and not the search bar in both Normal and InPrivate mode that is why I missed your question.
You are right when using Normal window, it automatically use the address bar but not when using InPrivate.
I believe it is best to report this issue from the link below in order to add this feature on Edge: https://techcommunity.microsoft.com/t5/discussi...
The feature of searching with other search engine was fixed last Mar 31, 2020 (check the spoiler from that link) however it looks like Microsoft has only done it with Normal mode and did not do the same with InPrivate mode. I will also report the same on my end but the more report they receive, the faster Microsoft will address the issue. I will also post to other forum so more user will report this issue.
Lee