on the other hand, I really want to implement features that allow players to pass down their characters' legacies to future generations by either having randomized, procedural-gen children or other family member, with the game world adapting to their actions.
The problem with that is just continuing with a legacy does not serve as a "Game Start" as it does not Resets things like your Progression and accumulated Resources, usually when you "Win the Game" you are already on top of the world having defeated everyone who opposes and impedes you.
One intresting thing to do is Simulate your winning "Empire" like 100 years into the future where your Empire has already Fallen into Decline based on the type of Ending you achieved, so a Diplomacy/Alliance style ending leads to Decadence while a Conquest Ending will have Rebellions rise up all over.
That's enough time for things to change and new mysteries to crop up to make for a new Game Start, it would also have the Charm of seeing the consequence and legacy of your actions and as well as retracing the steps to find out the mystery on what has gone wrong.
Of course you would need a pretty robust Simulation System and AI Director kind of thing to make that work properly, and it does not work that well if you want historical accuracy as this is more useful in a Medieval Stasis style fantasy world where Progress can be reset to some extent.
Alternatively what you can do as a New Game Start is to play against your Winning Empire as a new generation of Rebels. Like in Star Wars the "Empire" has "Won" so become the next generation of spunky "Rebels" that fight that.
Since you would know the Empire inside and out that means you would know what flaws and weaknesses it has that you can exploit.
You can't, because it's just not possible. There will always be a moment when you've seen everything so often that it become old. What make games like Skyrim or Fallout 4 to still be played a decade later isn't their sandbox approach, but their mods, that permit to put new variety into the game and then feel like you're playing something new, even if it's the same base game than ten years ago.
Depends. Gameplay can be it's own source of replayability as long as you can maintain the Challenge, there are 4X Games that are naturally procedural and randomized.
Even if you do the same Actions, and have the same Units,Resources and Progression how the game plays out can be diffrent and you might not necessarily win.
Especially nowadays with Challenge Ratings and Ascension Modes and other New Game+ Modifiers and whatnot.