Spectrum $\sigma(a)$ taken within the unitisation of a $C^*$-algebra

The mistake is to assume that the unitization process works fine for a unital algebra. It "works", but not as you seem to think it does.

Of course you can construct the unitization for unital $A$, but what you get is a different object. Similar to what you say, $(1_A,0)(0,1)=(1_A,0)$, showing that $(1_A,0)$ is not the unit of the unitization. And, as you say, $(1_A,0)$ is not invertible.

Whether $A$ is unital or not, the unitization is a new algebra, that contains $A$ as an essential ideal. In particular, when you start with unital $A$ the spectrum of $1_A$ changes as by construction all original elements of $A$ have zero in their spectrum as elements of $\tilde A$.