Ummm, no? Fresh (not yet degraded) hair and nails are Material Links, they were a part of the person's body, so they serve as the strongest links. They aren't sympathetic links, which is something described in Street Grimoire and can only be used if you have the proper metamagic (which allows you to use sympathetic links in place of material links for the requisite rituals.
Those rituals normally require a Material LInk, which is something that was a part of the person's body, allowing the ritual to target the subject by the bond of shared living material. That's why Material links are only viable for a short time, that bond degrades as the tissue being used as a material link dies off. Sympathetic Links, on the other hand, can be used through Psychometry, using the "psychic" link that a person makes with an object that they use over an extended period of time.
A focus was never part of the person's body, but you are correct that they do share the person's aura. But don't forget that a focus also has its own aura, which means it is still its own unique thing. That's why they can't be used as material links. A GM allowing them to be used as sympathetic links for a longer time than normal seems reasonable (normally sympathetic links are viable for double the amount of time that they were carried/used by the person). And of course, since they have the bonded person's astral signature, it can also be used for astral tracking like I mentioned before.
A better analogy would be like using crime scene evidence for matching to a person. Material Links are like DNA matching, while Sympathetic Links are like fingerprints. DNA matching is stronger, but more subject to degredation. Sympathetic Links generally last longer, but they can still get smudged and wear out. Its just that in the case of the Shadowrun setting, only specially trained people can use "fingerprints" but every mage can use a DNA sample and sick a ritual on whoever they happen to have. There's also Symbolic Linking (like voodoo dolls), but that's like using witness testimony, very finicky.