This axe has been passed down in my family for generations...

I mean... That's probably why it's passed on from generation to generation.
The new, cooler one is being used by daddy dwarf, while the dwarfling gets to keep the important family heirloom that would be too dishonourable to throw away or stuff in a chest somewhere...

"I'll never give up my grandfather's axe. But the handle does need replacing." Swaps out the handle with the one from the +1 axe. "You know, the head is looking a bit scuffed up too." Swaps the heads.
- Ah yes, the axe of grandpa Theseus.
- This old axe has had 17 new heads and 14 new handles in its time.

I can't help it my character's entire family lineage only managed to collect and pass down crappy first level items. Really, this is kind of on them.

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.
- There's a third party supplement I use for upgrading weapons called Ancestral Weapons. Basically, you get predetermined amounts of points to spend on upgrades that make the weapon better at certain things or entire certain abilities. As you level up the weapon by hitting predetermined milestones established by the player and DM (i.e. slay a high ranking follower of Asmodeus with Grandpa Paladin's gun), you gain more points and the levels of upgrades you can unlock get higher. It's fairly elegant overall and lets you shape the weapon as you go, rather than having to decide what everything can do up front.

This is why I prefer the paradigm of upgrading weapons, rather than just trading them in. An heirloom weapon with a history is soooo much cooler than just some random artifact harvested from a loot table.
You just provide the party with a magical smithing mechanic (flavor to your preference), and whenever they'd find a weapon, it's a smithy runestone or whatever, and they can use it to upgrade or change the properties of their existing weapon.

Technically speaking, whenever you pick up a new axe, that's +1 axe.
- Just tape them together for double the chopping power.

Time to Grandfather's Axe that. Take the +1 head and put it on your shaft.
(I know the +1 makes it magical, so you can't just 'destroy' the axe like that but shhh)

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.