If you are using Internet, Google, Stackoverflow at work, why insist that interviewee need to solve problems on their own?

To test their inherent thinking skills and base knowledge.

There's a huge difference between occasionally looking up something, and practically leaning on it. Ironically, the mass degradation of search engine result quality within the past ~decade has made it much harder for people to do the latter, and when they do, it shows much more clearly.