Is it true that radioactive elements contained within igneous rocks do not begin to decay until the magma from which these rocks were formed begins to cool? If so, why does the decay not begin until the magma cools?

Is it true that radioactive elements contained within igneous rocks do not begin to decay until the magma from which these rocks were formed begins to cool?

 

No.

 

However the decay products can be left behind in the rest of the magma when the minerals crystallise out so, at the time the crystal forms there might be (for example) uranium, but no lead (or, at least, very little lead).

Or Ar gas trapped in the crystal, in the case of K-Ar dating. The "clock" starts when the magma cools, not the decays.