One option is to simply make dying inevitable in the long run. Let's take Crusader Kings 2, for example. This is kind of a strategy game (the genre is hard to define) which takes place over several hundred years, so your character dying of old age (or sickness, assassination, in battle etc.) and getting replaced with their heir is a regular event in the game. When the player-character dies unintentionally, the player will often say "well, this was bound to happen sooner or later" and continue on with the next heir. A strategic suicide can be beneficial in some situations, but it can be quite difficult to do that, so most players won't bother unless they have a very good reason (like being stuck with a character with abysmal stats and traits).
Conclusion: Make it hard to die intentionally but inevitable to die in the long run, and the player will accept it when it happens.
Another way to make people treat the progeny system as more than just a cheap respawn system is to make sure that each consecutive character looks and plays different than the previous one. By giving the player customization options (but not enough to make them perfect clones of their parent), you can increase the emotional bond to each character and thus make the player reluctant to kill them off needlessly. By making them play different, the player will be compensated for the emotional loss by a new and fresh game experience, so their grief won't be as long.
When your game is story-heavy, you can also force the player to continue on with their next character by killing off their current character if and only if they reach a certain point in the story. When you want to have a plausible and coherent story spanning multiple decades, you will likely have to do this once in a while to skip time. The downside is that you will then have a fixed number of generations per playthrough and must make sure that every new character is properly balanced and playable for the next chapter of the story, so you might lose a lot of potential a progenicy system can offer. A mild example for this is Final Fantasy V where one character of the player's party dies in the course of the story and gets immediately replaced by his granddaughter. However, she inherits his whole character progression and is also otherwise almost mechanically identical to him, so the gameplay effect of this is practically zero.
And then there is the most extreme method to avoid savegame abuse: The roguelike concept. Don't have savegames. Save automatically when quitting but don't allow the player to make selective savegames. That way they are forced to accept the death of their character and continue with their heir.