They said that battleships were obsolete and they would never be used again either. Then they wheeled out the Missouri and Wisconsin for Iraq 1.
The battleships used in the Persian Gulf War were not used as primary fleet capital ships, they were used as shore bombardment vessels, at best a secondary capital ship role. Additionally, no one has bothered making any battleships for the past ~70 years; at least for the moment, the battleship is dead as a primary capital ship, and most likely dead as a monitor. The investment is just too high for what you get.
The reason why I feel beams are the obvious winner in terms of accuracy, and I will use an FPS term here, is hitscan. If you've played quake or a similar deathmatch type FPS, hitscan is what a railgun is in those games. You point, you click, you hit. A railgun isn't going to be hitscan at hundreds of kilometers, or even thousands. However beam/lazer IS hitscan up to a point. You aim, you fire, you hit. This is virtually a simultaneous action.
It depends on the range of the engagement and the speed at which the participants move relative to one another. Sure, lasers are essentially guaranteed to hit, assuming that you're at a sufficiently short range and that your weapons are pointing in the right direction ... but have you fulfilled those requirements? You claim that lasers are the best at short range, but fail to define short range.
Self guided railgun munitions, but not self propelled will also be a thing.
Given that in space there is no atmosphere and very little else which control surfaces or other guidance systems could work upon, I find this assertion highly questionable. Gyros are good for stabilizing and for rotating a body, but not for moving it in the absence of something else to act upon the body. You need to have thrusters of some form or another on the projectile if you want to do anything beyond stabilization, and once you put thrusters on the projectile, there is very little to distinguish it from a self-propelled weapon except quantity and quality of thrusters.
Prediction is the effective limit on a kinetic weapons range. However, this assumes that the enemy can predict where you are shooting. In a way this becomes part of the game of long range fleet engagements with kinetic weapons. You don't aim at the target, you aim where it can potentially be at the time you expect your rounds to hit. Not exactly game breaking concept. However, given dozens or more weapons, you don't HAVE to aim them all at the exact same spot. You can aim long to account for acceleration, you can aim short to account for deceleration, long and high, long and low, short and high, short and low. You you probably don't have the guns to account for every variable, but this is a way you can maximize your chances of getting hits at ranges where the projectile could take minutes to reach the target.
Have you ever looked at a plot of potential end positions given X time, Y maneuvering capability, and Z initial conditions? It gets messy fast, and small errors in initial conditions or the assumptions about maneuvering capability can lead to significant errors in the predicted location. Predicting the position of a target well enough to hit it at long range is not a simple problem.
Also, do you not understand what is meant by "how accurately I can predict the future location of the target?" You speak as though you're revealing something I overlooked; what you describe is an attempt to predict the location of the target, coupled with an attempt to counteract the inaccuracies by spreading out your fire a bit.
Furthermore, there is at least one situation under which target prediction is not necessarily the limiting constraint on my weapon ranges, and that situation is when I cannot aim my guns with sufficient accuracy to hit the target. Even a small aiming error can result in a miss of a considerable margin. Maybe in Perfectland you can have guns that align to the exact bearing you give them at the instant you give them the bearing, but real components take time to orient themselves and don't end up in exactly the correct spot. The farther off the thing you're trying to hit is, the better you need the actual orientation of the weapon to align with the real orientation. On top of that, you're not only dealing with the target's location. Your own ship is likely performing maneuvers of its own, and possibly suffering hits, which can and will throw off the aim of the weapons; your ship also has to absorb the recoil of the shots it fires, which will create vibrations and modify the ship's position from what had been expected. Gunlaying accuracy creates a range constraint which should not be neglected. Yes, 573 km (against a 1 km target) sounds like a fairly decent weapon range, but that was for a weapon that had a gunlaying error of no more than 5 hundredths of a degree. You think your guns are going to have this when the ship's in combat? Despite engaging in whatever maneuvers it needs to, despite the vibrations from the other guns being fired, despite the hits the hull has to absorb? Maybe it will, but I'd tend to suggest you're being a bit optimistic.
Also, how fast is the projectile? 573 km is basically nothing to something that travels at 0.5c or so; it'll cross that distance in a bit less than four thousandths of a second. If 573 km is where your gunlaying error starts to interfere with your ability to hit the target, how much do you think your ability to predict the target location matters, for a weapon which travels at 0.5c? With that +/- 0.05 degree aiming error given earlier, I have a 99.7% chance of sending the shot through a circle 1 km in diameter. What does that translate to for a target ~500m long, such as, I don't know, the T.A.S. Crusader? Let's assume a uniform distribution inside the 1km-diameter circle since it's easier to deal with (every point on the circle is equally likely to have a shot pass through it) and let's assume the T.A.S. Crusader has a circular profile of diameter 500m. Since it's a uniform distribution, the chance of hitting the target is equal to the ratio of the area taken up by the target to the area through which our shots can pass, or one fourth. So with a 0.5c railgun at 573km with an aiming error of 0.05 degrees, we have about a 25% chance of hitting a 500m long target, despite it only taking four thousandths of a second for our projectile to travel far enough to hit the target. This is less than ideal, but might be acceptable. If you double the range, you double the area that our shots might pass through, but the target's size is unaffected and distance into the future that we need to predict the target location is now a whole ~8 thousandths of a second, but we're already down to a 1 in 8 chance of hitting the target. Still convinced that your ability to predict the target's location is definitely going to be the limit on your ability to hit the target?