The mental monologue debate is just another case of "what if the red I see isn't the red you see?" There's just no way for you or your friend to prove anything to each other. It's a fun conversation, but you're not going to get any answers.

The question of whether someone "has" a mental monologue is also complex. Is your friend saying they don't usually talk to themselves when solving problems, or that they never do? Are they saying they can't talk to themselves? If they don't do it often, and you do it more often out of habit, that's a different thing than "not having a mental monologue". The problem with these articles is that people conflate all of these concepts.

Also, a lot of data supports the idea that people aren't very good authorities on how their own minds work. Decades of research on psychology and consciousness can attest to that.