i've never played ebberon but i don't see any to them the reason WHY you can't consider them just some generic construct species? it's not like they're tied to some god or religion or whatever and even if they are it's not like that kind of lore doesn't get ignored in basically every other species,
i've listened to a podcast that used them as just that (the generic construct species) and yet there was absolutely no wrinkles from it not being ebberon, just they exist because a wizard discovered how to put souls into bodies of wood and metal, done deal, 'artificial person' is possibly one of the most common concepts you could come up with for a species.
Warforged were designed with a specific world assumption in mind. They were created en masse in a society that can make sentient golems on a production line. Their name speaks to their purpose as soldiers. They have no means of procreation due to all the official forges being shut down (and only a few illegal forges secretly in production) and the oldest of them are 30 years old with no idea how old they live. They have a single generic form that some have modified, no concept of gender, and no community of their own. All of that informs them from the most basic level.
Lots of DMs use them as generic PC golems (a role better served by autognomes and reborn) but that isn't being a warforged, that's just using their stats for a different concept. They aren't warforged the same way an elf is basically an elf. Those golem races lack the creation forges, the purpose as a soldier, their status as a recently freed people without a culture beyond soldiering. They may use the same stats as warforged, but they aren't
warforged the same way a Faerun dwarf is the same as an Oerth dwarf.
In short, the story of warforged is tied to Eberron and to remove them from that is to make them something different. You could not put warforged in the PHB and expect them to function the same in Faerun, Krynn or Ravenloft. That is in sharp contrast to elf, dwarf or such where they fill the same general role in every world, but get mired down in minutia of the difference between a valley elf, a sun elf, or a Qualinesti (pop quiz, can anyone tell me the difference between these three elf cultures without looking it up?)
What I think would do D&S a lot of good is for it to focus on one of those types of elves as the default lore instead of trying to accommodate all of those and more. But D&D is a Multiverse and therefore the info on them is a mile wide and inch deep, so we're stuck with "magic elf" "tree elf" "water elf" "underground elf" as being as deep as the lore will go on them.