This is a "Wild Mass Guess" entry, where we pull out all the sanity stops on theorizing. The regular entry on this topic is elsewhere. Please see this programme note.
Integrated circuits are LosTech, and all computers are analog.
This would explain why an additional targeting computer takes up several tons and a lot of hull space. It might also explain the Short-Range Long-Range Weapon thing.
Despite the name, there's actually a lot more to a functional targeting computer than just a bunch of chips to be plugged into an existing system as-is. It's all the extra peripheral performance-enhancing gear that takes up the tonnage and space (and yes, that's canon).
Indeed, such things as additional recoil/motion compensators would take up a fair bit of room and tonnage. Additional sensors attatched to the weapons allso adds more tonnage/crits, though for ease of gameplay/design, all those bits were lumped in with the actual computer.
It's also heavy and bulky for game balance reasons. Game balance explains a lot of the other weirdness in the game. Remember the Autocannon/20? The book establishes that an AC/20 is somewhere in the 185mm to 200mm range, and yet, its maximum range is less than 300 yards. Guns of that size exist in real life... an example would be the MK-71 naval 8-inch gun (200mm is just shy of 8 inches). The Mk-71 has an effective range of about 25 MILES, about 150 times the range of an AC/20, but as it stands, the AC/20 can already destroy most Light Mechs with one or two shots. If you combined that kind of damage with a realistic range, game balance would go out the window.
I've always considered it less of an absolute range problem and more of an accuracy problem. Mechs are big, but compare the gun/space ratio of a mech's arm or torso, then look at the gun/body ratio for a ship; the ship has a lot more room for recoil compensation and such. It's not that they can't fire far; it's that they can't fire far and do so accurately.
The short range of the larger A Cs probably can be explained by low projectile speed. If it was higher, the 'mechs wouldn't be able to handle the recoil.
Adding to the previous note, each hex on the map is 30 meters across. If you used the REAL range of guns that size, the map would have to be close to 1,500 hexes long. You'd need a Segway to get from one end of the map to the other.
Also in one of the more recent rulebooks (not sure which, mind) there's a note from the designers that for the sake of effective gameplay, the ranges in game are much shorter than what they would be in the fluff.
This point seems to have been addressed in a very roundabout way in Alpha Strike, where units are given two more range bands in expanded rules — Extreme (21-30 hexes, or 42-60 inches), and Horizon, the latter being effectively Exactly What It Says on the Tin. However both of those have insanely high to-hit modifiers, meaning that even an ace pilot would need a good bit of luck to actually hit anything at Horizon Range. It's a +8 modifier, modified by movement and terrain. It'd be insanely easy for even a skill 0 pilot to need a 12 or above on 2d6 to hit. Also, it's only weapons already capable of actually dealing damage at long range that can qualify, so the 200mm AC-20 mentioned above still is restricted to 300 yards...