Until the 11th Century or so in Europe, good swords could be very expensive. They were generally made from layered or twisted steel, and pattern welded from different core and edge components to get a blade that was both resilient and able to take a sharp edge. This involved a lot of labour and expense, and it made a lot of sense to pass these heirloom weapons down through the generations.

The British Museum has a lovely replica of an early medieval sword from the Sutton Hoo ship burial, in which you can see the remarkable blade craftsmanship. These were expensive objects, and you would have to be quite wealthy to discard them or let them languish unused while you had sons who needed arming.

Things changed by the High Middle Ages. Advances in steel smelting reduced the need for such expensive and time-consuming sword production methods. Advances in armour development also led to changes in sword design that meant heirloom weapons, whatever their quality, would have been perceived as antiquated and unsuited to the contemporary battlefield. For example, by the late 13th Century, swords were increasingly designed with steep thrusting points, whereas on earlier weapons the points were largely an afterthought, since you generally struck with the edge. It was possible to rework an old sword to bring it up to modern standards, but only to a small degree. Too much reshaping, and you would remove too much hard jacket steel and expose softer core steel, and the blade would be permanently ruined. So those beautiful and expensive early medieval swords that would have been great heirloom weapons did not really have the long-term technological relevance to continue that tradition past the 13th Century or so.

Interestingly, Japan never experienced Europe's steel-smelting revolution, so the traditional pattern-welding construction techniques persisted right into the modern (meaning post-medieval) era. The military stasis brought about by unification in the Tokugawa period also meant that swords did not experience strong evolutionary pressures away from those traditional designs, so older weapons experienced fewer and less radical attempts to rework them into newer styles. (There was a significant movement to rework traditional samurai swords into neoclassical sabre mountings in the late 19th Century, however.) The heirloom qualities of swords thus remained strong in Japanese culture, and there are many surviving samurai swords whose provenance can be reliably traced back to medieval times.

I can't find the quotation now but for my Masters research there is an Anglo-Norman text (late12th/13th century) where Anglo-Norman aristocrats are asked to prove their heritage. One family brought out an old rusted sword that was used at the Battle of Hastings. So while it was passed down, it certainly wasn't reguarly used.