PAGAU said:
'High thermal conductivity' means the Silver bar absorbs the cold quickly in the fridge which is good when you want an instant ice-brick.
High thermal conductivity also means that the cold bar will absorb the heat of your forehead very quickly/efficiently too. In other words, yes, an excellent instant ice brick for your headache although best to have a second bar in the fridge and rotate.
lets use a margin value as to make clear what I said:
imagine a material with an infinite heat conductivity.
you put it in the fridge, it's instantly the same temperature as in the fridge.
you take it out the fridge, it's instantly the room temperature, and by the time you put it against your forehead, you could aswell slap your head for no reason.
abit away from this margin value: you only have a minute of 'cold'.
abit further away: you have 2 minutes.
This makes clear that a higher heat conductivity doesn't make it better / more efficient as a 'cold pack'. Like alot applications, it's a trade off, and using silver instead of water in a cold pack just inflicts you more trips to the fridge, as a net result pro/cons.
For the same reason, it's exactly the opposite as your last sentence, see the margin case example: with an infinite heat conductivity, it's completely useless to have a second bar in the fridge. Put the first ('used') one in the fridge, and it's instantly cold. Why then having a second?