I've hired tons of people, I've never offered someone less because they came in via a recruiter. You typically budget for salary + recruiter fee and if someone is hired organically then that's a bonus.

You don't pay someone less because they came via a recruiter because then they are going to be poached for being underpaid. You also don't pay someone more because they didn't come in via a recruiter otherwise now I have guys working on varying salaries which creates issues if that comes out in a team.

source: prior founder, hiring manager at large companies, etc

You likely just didn’t realize it. Usually the head-hunter cut is a % of the new hire’s salary for a length of time and is paid by the hiring company. So you never see it... but if someone gets hired directly then they can usually negotiate a higher salary instead of that money going to the non-existent headhunter. Another way of saying it is that initial offerings through a headhunter are lower since the cost is higher to the company.

This is my experience, either the recruiter will want to pay you directly, thus the company can avoid its own HR red-tape, or their deal with the recruiter is deliberately hidden from you, but you still join FTE. Of course, there's a million different ways to do this, but an organization like Triplebyte is upfront about taking 30% of your salary for two years.

If the hiring officer has a budget of 150k for the first year, and the recruiter charges 30k, how much are they going to offer you?

You can still negotiate to what you're worth. If you're no better than the applicants they got without recruiters then good luck, but that's why companies are hiring from recruiters, they can't find the talent on their own.