It's seems that Rockstar originally tried to add another melee weapon for GTA 4, like the tonfa for the cops (thanks to some unused dialogues from the cops using aparently tonfas), which is in this case a pool cue, before that TLAD added it exclusively for that DLC. But unlike the pool cue from TLAD, which it's actually a half of a pool cue, the first version for GTA 4 was going to be a full pool cue, like the one from GTA SA. It is unknown why Rockstar scratched the idea of having a full pool cue as another melee weapon for GTA 4 and probably for both TLAD and TBOGT. It is also unknown where this weapon was suppose to spawn in the game, but it's very possible that it was going to spawn on the pool table from the Jamaican bar in East Island, after playing one game at least, just like in TLAD but with the half pool cue.
Funny enough, you can actually spawn this weapon in GTA 4 & EFLC using a trainer, but sadly it doesn't work properly. The weapon doesn't crash the game or anything like that, but when you try to hit some NPC or any object in the game, the protagonists don't hit them at all. They just staying in a some kind of "fighting" animations when they didn't even hit anyone or anything at all. Luckly, I managed to fix the pool cue's script in the WeaponInfo.xml from GTA 4. Believe it or not, it was not so difficult to fix, I just needed to support myself with the baseball bat's script (which it was very similar like the unfinished scripted for the full pool cue).
After fixing the pool cue's script, I really expected that the weapon was going be a complete copied-pasted baseball bat script, but surprinsingly the weapon have its own sound effects when you hit someone or something, and even a few exclusive animations for the weapon. Aparently, Rockstar had that weapon almost finished (like a 90% complete, but they just needed to fix the script for the weapon), but I'm guessing that they needed to finish more important stuff, before the game's release, and then they completely forgot about that weapon. A possible clue about this especulation of mine is that the weapon itself have a very LQ textures and even weapon icon, comparing it with the rest of the weapons in the game, like if that pool cue was in an early development, but finished in animations and sound effects (except with the script), and then they stratched the idea to focuse in another stuff for the game.
A very ironic thing is that Rockstar made later another unused full pool cue for TLAD, like the first one in GTA 4, but with a much better HQ model, textures and weapon icon, but also with no script. Then I thought "Why I don't replace this LQ pool cue with the unused HQ full pool cue from the TLAD, now that I managed to fix its script?". The result was actually a success, the weapon feels very official in the game like if it was from Rockstar itself, and you can use this weapon not only in GTA 4 but also in TLAD and the TBOGT.
THIS LAST PARAGRAPH IS VERY IMPORTANT TO READ. In case that you have installed my "Max Ammo like GTA 5" mod for GTA 4 & EFLC and you're wondering "Is this mod going to change also the ammo script for the weapons to the default one?" or in case that you want the full pool cue melee weapon fixed but with the default ammo script for the weapons, don't worry, I made two versions of the WeaponInfo.xml. One with the vanilla ammo script, and another with my "Max Ammo" mod.