Author here. The main reason not to share is that if you share at the beginning of the process, they will inevitably ask you how things are going with Company X, Y, and Z. Let's say you get rejected from a few of them. It's going to be increasingly hard to hide that, and now you lost leverage. Better to share when you control the flow of information.

I wrote a bunch about why sharing doesn't make sense and how to avoid it, while still managing your timeline: https://interviewing.io/blog/sabotage-salary-negotiation-bef...