...and stock traders: an industrious person could probably figure out how to do a minimally adequate job of all these things assuming the barest, simplest case in the world. There are three issues: competency in the face of the range of issues that commonly come up in these fields; what it does to negotiating; and the social-impact difference between people doing things for themselves and third parties doing things for hire.
It's true that any idiot could try to sell their house. Just put an ad online, stick a sign in front of the house. But suppose Joe Schmoe neglects to tell the buyer that the basement floods every time it rains, figuring, what they don't know won't hurt anyone. A few months after closing they are served with a lawsuit.
Joe Schmoe signs off on the contract and then asks the buyer's agent when he'll get his check and if they can stay for 3 more months, since the wife is pregnant. Sorry--the contract says closing in 3 weeks, possession on day of closing. As for documents the seller has to provide at closing, hopefully Joe will figure that out.
In practice, all sorts of complications, usually minor but sometimes major, pop up. For the average joe schmoe, this is new territory; for a licensed broker, and especially one who's experienced, it's not.
So, in the first place, there are technical issues that Joe Schmoe is not equipped for, and that could cause problems getting to closing.
Second, it is generally better to negotiate through an agent than on your own behalf, unless you have some experience doing that. The problem is that home sellers (and buyers) let emotional issues cloud their judgement. What might seem like a personal affront or an insuperable obstacle to doing a deal is something that your broker could see a solution to, because they'd experienced it before. For many people, the mechanics of showing your property can be time-consuming and burdensome. And if you don't offer a competitive level of cooperative commission, you'll probably miss out on a number of buyers who are represented.
Finally, the social impact. If there were no licensing requirements, any Joe Schmoe could print business cards and call himself a real estate broker. Con artists and fraudsters would crowd into the field (go back to around the year 1900 or so and compare the security trading field). The activities of brokers have a wider impact than just your transaction; they might work on a couple of dozen a year. An influx of unlicensed people would just make things harder for consumers. There are already enough cases of licensees being disciplined for not serving their clients the right way. If you removed licensing, most of that discipline-worthy activity would go undeterred, and would almost certainly go up.