> why not at least try giving them a number and give them a chance and see if they would beat it?

If it's obvious that the recruiter is uncooperative or that the role isn't prestigious/top-tier, this is exactly what one should do.

The reaction to blowout numbers may just surprise you, and perhaps not for the better..

In my life I've been more upset by a lack of resistance in negotiations (of any kind) than immediate supplication, because it invariably means: "I could've done better, my appraisal was wrong"