> In any large company, the team you're negotiating comp with is almost entirely unrelated to the team you'll actually be working with

I've a few data points on this, and disagree. Hiring Managers (and their managers) have to approve and signoff on things, and VPs are needed for major deviations from the norm. Yes, the HR team can do many things but it's all done in conjunction with the hiring manager.

I distinctly recall talking with one of my GM's one day about a new start. The discussion was, "I'm excited for $person to start, but I'm really worried (s)he's a player of games. The negotiation process was non-stop angle-shooting, and please keep an eye open for actual work vs politics."