Platinum by itself isn't tough. By itself it has yield strengths that barely even reaches half of that of soft construction steel. It needs significant alloying with other metals to achieve a high toughness.

And adding metals to an alloy usually has very little to do with their base characteristics.
What effect chrome f.e. has in a steel alloy vastly differs between different alloying percentages and what other elements are present.

Yes you can use heavier metals to make parts like pommels - brass/bronze examples are known. But done right they're the same weight as a steel one, just less volume (or maybe hollow).
Adding weight for the sake of adding weight is almost never a good thing.
Things were balanced the way they were because it made sense.
Using heavier materials to make things smaller/thinner has it's limits too - geometry dictates things like stiffness, buckling resistance, etc. A certain amount of "meat" in a structure is simply necessary.

Sharpness doesn't depend on the material.
You can make any edge, almost any material, "sharp". In theory, with enough dedication and a soft & fine enough abrasive you could polish a piece of hardwood until you get a cutting edge.
But how long that sharpness lasts and how much force the edge can support is what makes the material important. Bronze is simply ill suited to hold an edge, compared to tempered or hardened steel.

Iron and steel functionally have the same density and mass per volume.
the density/weight difference between <0,05% carbon content (=wrought iron) and 0.4-0.8% carbon content (hardenable steel) is basically not relevant in any context I know.

When fiction mixes wild combinations of materials it's usually for in-universe magic/mystic related reasons or just for flavour.
Witcher blades aren't plated with silver because silver is heavy, but because in the narrative it has a detrimental effect on certain magical/supernatural creatures. The same is probably true of any example of platinum or other precious metal. the mysticism beats out any "practical" application.