Electronic – 3×2 solder pad connector

The best way is to desolder the old cables from the scale's circuit board, and solder on the new ones. That way there are no messy splices. However, to do that easily you may need a solder pump (cost: about $5-$10 USD) to vacuum away the solder from the existing joints. Without a solder pump, you can desolder a through-hole-mounted wire simply by pulling it away while the solder is liquid, but solder may clog the hole. To get around this, you can heat the solder that remains in the hole while inserting the new wire; if you can't get at it with the tip of the iron, then simply add more solder to the area.

The next choice after that is to make rat tail joints (stripped ends of wires twisted together) between the new and existing cable (or else Western Union splice which is neater), and then strengthen these joints with solder and perhaps cover with electrical tape or hot glue. The Western Union is not well-suited for braided wire, but this is only true of the splice alone, not if the splice is flooded with solder.

@Kaz & @LongStrokinYerMomma are near to the right explanation.

When you talk about mechanical properties of a metal/alloy we have to consider lattice structures. And in this case not much chemical reaction is of our concern.

You see, two phenomena are responsible for this observation:

1. Re-crystallization

Ability of metal/alloy to be drawn into wire is called ductility. When solder wire's billet is drawn through various dies of reducing diameters - it undergoes a process called strain hardening which makes it more resilient ^{(i.e. to bend repeatedly without fracturing easily)} to shear/deforming forces as compared to the initial cubical billet of same alloy. Hence when you melt it, it loses strain hardening & undergoes re-crystallization which makes it appear more brittle.

2. Perfection of lattice structure

Diamond is the hardest material, not just because of its bonds but because of its perfect lattice structure. If you compare degree of perfection of lattices PER unit mass of a small cube, say 1mm³ & a large cube, say 20mm³ of chemically identical alloy/metal/mix, you will find the smaller cube to be more perfect hence stronger/harder than the larger cube, even though their chemical compositions are exactly same _{(this is what user @LongStrokinYerMomma pointed out in his Abstract from that paper)}

To get a simpler everyday feel of it, think of breaking a stick, you can easily break a 2-feet long stick but not 10-cm long stick, yes in this case lever-action/torque-arm plays a role, but you get the idea.

Your logic:

The solder wire is less dense, either through being hollow or having a flux core, making it seem easier to bend. This seems less likely, because tin whiskers seems much harder than solder wire despite being thinner.

is perfectly valid, it partly explains why soldering wire is pliable. But note that the assertion Solder on a circuit board is just as soft as the wire solder it came from is definitely incorrect.