I'd think fee only figures into the salary conversation when fees are exorbitant (which they can be), otherwise it's a cost of doing business for some companies. Once a startup taps out their friends and family plan, they often reach out to me for some hires, and those usually lead to some referrals from those new hires and the company can subsist off friends/family of the new hires for a little longer.
This is part of the reason I charge lower and fixed (fees paid in advance, well below market rate, and fixed meaning not tied to salary) fees than my competitors, as it takes this argument off the table from both sides. It doesn't prohibit a company from making a hire because the 'tax' on the hire isn't high enough to be prohibitive, and candidates know that the 'tax' on their hire is also relatively low and prepaid. Under this model, any hire is judged based on compensation to value, and not based on a high 'tax'/fee.
Edit: a specific scenario from earlier this year. I was cold-contacted on LinkedIn, the job sounded interesting, so I talked with the recruiter, ended up applying, interviewing, and declined an offer. I told them the range I wanted, they asked what I was making. I dodged the question, so they asked if it was more-or-less X, and I confirmed. Turns out that when they talked to the company, they said I had wanted about 10k less than I had actually said I wanted. The number they said I wanted was a 10% bump on my current salary, and it seemed to me that they were trying to find out the lowest number they could give that would still be a 10% raise. It seemed pretty scummy to me, and I'm not going to work with him again.
There is a ton of 'fear' or at least awareness about being at some disadvantage if a company knows your current salary. I personally think the only number that matters is your salary expectations.
If you are truly underpaid and you are aware of this, then your expectations might seem a bit high relative to where you are now, but not high if you consider the market. As an example, when candidates move to the US from foreign countries, they can see themselves earning double or even ten times what they made at home. If you make $10K US in India and tell the recruiter that, and the recruiter shares that info with the hiring company, do you really think the company is using that number as some baseline? Of course not. Expectation is the key number here.
If you are overpaid for a certain market (which can be dangerous from a career mobility perspective) and aware of it, usually due to some externality, your current salary makes little difference. When someone changes geographic markets, such as moving into or out of SF, their current compensation is basically thrown out the window. Should you have fear telling a recruiter that you were making 100K in Philadelphia when you are moving to SF? Your market value in SF may be closer to 150K, so current salary means little.
If you know your market value - and a good recruiter who knows a specific market should be able to price you out - discussing current salary doesn't matter much. When I'm working with underpaid candidates, I'll always mention to my client company that my candidate is underpaid and we expect that will be remedied during this job search.