If you've got enough details about the company and role to contact them directly (and you haven't engaged with the recruiter at all) then you aren't doing anything wrong by contacting them directly. They may still choose to have you go through the recruiter but that will depend on what agreement the compnay has with the recruiter and possibly how they prefer to go about the hiring process.
If you have engaged with the recruiter at all then it gets murkier - many will ask you to agree to exclusivitiy when they start the process meaning you could encounter difficulties when going around them. Also at the very least it can look underhanded. So once you've started talking to a recruiter about a vacancy going around them isn't a good idea unless you are encountering significant incompetence from them - and ideally in such a way that you can substantiate with the company to prove you aren't just dropping them now you have what you need.
As to why recruiters exist there are various reasons depending on which side you are looking at:
Finding candidates and the initial filtering steps is time consuming if you aren't already geared up for it. Recruiters and agencies have databases they keep of CVs and contact details that would be inefficent for an employer to build and maintain unless they were recruiting all the time and hunting down candidates like this would be time spent by their existing employees on tasks other than their day job so there's an indirect cost to them there.
Which of course is why they aren't just "saving" money on recruitment which they could pass directly on to your wage and since many recruiters work on the basis of a cut of your first year's wages they are only paying that cost once - an increased wage to the candidate has to be paid on an ongoing basis. To use your figures if the cost of the recruiter is $1.5k then giving the candidate even an extra $500 a year in higher wages will work out more expensive then the recruiter in just four years.
Similarly hunting down openings and the right contacts etc is time consuming for a candidate, and if they are already employed they likely have a limited amount of time to spare on the job search, and that's even assuming they are actively looking! A good candidate who is employed and not looking might be tempted away by an attractive-sounding position dropping in to their inbox but they aren't going to even know about it were it not for the recruiter putting it out there.
Similarly when it comes to negotiating wages the recruiter is motivated to get you the best possible wage when their fee is a percentage (obviously some work on fixed fee so this won't apply in that case) so it is in their best interests to fight as hard as they can to a) get you the job and b) get you the most money. Yes you might be able to do better, negotiating skill varies from person to person and you might well be awesome at it and you are certainly just as motivated as they are to get it right. But realistically the recruiter is doing this day-in, day-out so they are getting more practice, are more likely to be on top of the market conditions and the rates a given role or skill set should get etc. So the majority of the time they are probably going to be in a better position to maximize the wage then the average Joe Candidate who goes through this process once every couple of years.