Very cool idea. However, there are some problems that need to be worked out. First and foremost, True Elements are very volatile, and very unstable. Also, just being enchanted does not give the weapon the ability to withstand blows like a sword of hardened steel does, for instance. The reason alloys exist is because there are many, many, many materials that need to be mixed out of their 'pure' form to be hard enough to actually be useful. This is why you don't see pure gold anything. Gold is soft, to the point where if you made a ring out of pure gold, you could easily crush it in your hand. Even Orichalcum, which is more common and easier to work with, is still alloyed with things to keep it from being destroyed.
Second, True Elements are not benign. They don't pick friend or foe. True Fire in its pure state will burn anything and anyone it touches, or worse.
Third, note that the value of True Fire is 1.5 MILLION nuyen per unit. Doesn't say what the unit is, but we can be sure that you'd need more than one for anything more than a Swiss Army Knife, yeah? Even if you source the material yourself, and manage to keep it from breaking down before using it in a focus, you've still got millions of nuyen on hand. That IS going to catch someone's notice.
Fourth, a sword made entirely of fire, without a spirit riding in it or a spell effect sustaining it, is GOING to make news. The moment you use this thing, your Public Awareness, if nothing else, is going to start shooting up, because you've got a fragging FIRE SWORD.
But I'm not here with just doom and gloom. You can use True Elements in the making of a focus or fetish, but they won't be ENTIRELY that element. You can use up to two units of True Elements in a focus, but the effects are listed on the bottom of Page 30 of Parageology, namely, you get to pick one of four effects from a list for each unit of True Element you use. One of those effects is the ability to manifest the elemental effect at will (and it won't harm the person it is bonded to). That means if you put True Fire in a sword weapon focus, you can have it manifest fire on the blade. In game terms, this would likely convert the damage to (fire), including the secondary effects, but the stats would otherwise be unchanged.
Or, for the same effect you can use the Fire Aura spell, get an ally spirit to have a form of a flaming sword, or somehow convince a Spirit of Fire to inhabit a sword.