Which is why, when you're a good DM, you also create a homebrew upgrading material so that your player making good roleplaying and rp combat with his family weapon keeps it and upgrades it throught the adventure.
For players that do this backstory idea : when you find a new better weapon, ask your DM if it would be possible to use parts of it on your own weapon for RP reasons.
There is no downside to this. Unless you count destroying a weapon. But normal ass weapons's value is not IMO good enough to stop the idea entirely.
Technically speaking, whenever you pick up a new axe, that's +1 axe.
Just tape them together for double the chopping power.
This is why PF2e made weapon and armour runes transferrable. It also mitigates the problem of getting a powerful magic weapon that nobody can use - just move the runes onto a weapon you're already using.
"Hang on a moment! This isn't my grandfather's axe; that one is!"
