I meant that if Microsoft launched a privacy-focused product and hid their involvement with it they would receive extremely negative publicity on the sort of websites people who use privacy-focused products read.

Completely leaving aside the harm that would cause to the Microsoft brand, it would also be completely useless. Essentially no one would switch from DDG to Microsoft's (hypothetical) shady clone.

Your hotel examples aren't relevant because people searching for hotels and people searching for private search engines don't evaluate details about corporate ownership the same.