I've heard this "don't name a number first" advice so many times, but personally, it never seems to work. My current comp is significantly higher compared to others with my level of experience. So when I refuse to give a number, the recruiter ends up guessing a number that is significantly lower, and making a lowball offer. I've also had many other experiences where after many hours of interviews, I find out that the…
The other point I have is that the first number isn't all that important. Yes, it might anchor the starting point, but you'll be in a stronger negotiating position when the written offer is actually made.
I just went through this recently. Yes, their hiring manager asked me what my requirement was for a salary and I gave them a number without any fuss. In their first formal offer, they gave me the number I wanted; however, by that point in the hiring process, it was clear that I was a top candidate, and with other competing offers, I was able to get them to come up another 30%.
In the end, I don't think the best approach is to force the company to make the first offer. They aren't going to give their best offer if you force them to make the first move. The only time that strategy might make sense is if your best number is much lower than their worst number; in that case, you did a lousy job in the research phase and need to take a step back.