Third base coach is definitely a plus. He can stop a runner at third or wave him on. He is the eyes of the runner.
First base guy? He might argue a close play , but mostly he shares chit chat with the runner and maybe a high five for running out a hit.
So, what’s his official purpose?
Tells the runner whether to stop at first or keep on going. Or to run through or make the turn and hang tight. He’d also help strategize a steal.
He’s also in charge of collecting a runner’s various batting gloves, shin guards, elbow protectors and whatever do-dads he needed while batting.
As ZipperJJ mentioned, he tells the guy weather to stop on first or keep running. The runner can go much faster if they’re not trying to track the ball while they’re running. That could be the difference between a double or just a single.
While there is some benefit to having a first-base coach, their role is not nearly as important as that of 3rd-base coach. IMO their on-field position is mainly ceremonial: The position will either give a younger management candidate some on-field exposure (all 1st base coaches have other duties such as outfield/infield/baserunning instructor), or provide clubs a way to get retired fan-favorites onto the field (Rickey Henderson’s recent hire as the Mets 1B coach is a typical example, and Billy Williams was the Cubs 1B coach for years until he took a front-office job). I don’t even think 1B coaches commonly relay signs to the batter, though I wonder if this might vary from team to team depending on the position of the dugouts (I doubt it).
In some minor/non-affiliated leagues, where the number of coaches is particularly limited, the manager himself will commonly serve as the 3rd base coach, illustrating the importance of that on-field position over the 1st base coach.
Frequently, the first base coach will keep a stopwatch, and time the pitcher’s delivery to home. For a basestealer, this may indicate whether this is someone to run on or not. Also, provides a nice symmetry to the 3rd base coach on the other side of the diamond.
Of course, you have to bear in mind that first base coaches can be extremely important below the major league level. The OP didn’t specify the major leagues as the limit of the discussion. In amateur ball, high school and college ball, or recreational baseball and softball variations, the runners often DO need reminding of the situation and appropriate responses, or to be told whether or not to advance to second. Major league teams have first base coaches largely because there’s coaches at first base all the way up the ladder, so you may as well have them there, too.
They also remind the batter of the situation even in the majors: “It’s two outs, now.” While this seems trivial, it acts as a backup to make sure the player knows the game situation. Even pros can have mental lapses, and this helps prevent them.
The coach will also shout something if a pickoff is being attempted, again, to make sure the runner is on top of the situation. Similarly, if the first baseman misses the pickoff throw, the coach will tell the runner to advance, since the runner may not be able to see the play. He can also warn the runner of the hidden ball play.
He is definitely not as important as the third base coach, but does have a role to play in the game.
Or told when to come back.
I was acting as first base coach for my team in our city league softball game yesterday. The bases were loaded, and after the ball was hit to third base there was a play at home plate. The catcher dropped the ball, and the hitter, who had just arrived at first base, decided to try and make it to second. While running hard, he completely failed to notice that the runner on second was not advancing to third, and i yelled at him to return to first. He was half-way to second by the time he turned around; luckily the other team wasn’t quick enough to pick him off.
As manager of my son’s little league, I tell my players to listen to the first-base coach as to whether or a hit was a foul ball. With the umpire behind them and everyone yelling, it help them to have one person to listen to.
In the minor leagues, the first base coaching job is usually performed by a player who isn’t in the lineup, with the manager serving as third base coach. I think this is also a common practice at both college and high school levels.
Slight hijack, since the question appears answered- assuming there was video of this, did any TV channels air the video?