In the renormalization of the QED, why do we need superficial degree of divergence?

Question

What is the intuition behind superficial degree of divergence and why do we even use it at all, consider there are numerous exceptions involved if we use this concept?

Answer 1

The main point of superficial degree of (UV) divergence $D$ of a Feynman diagram is to help pin-point possible (UV) singularities.

Since every connected Feynman diagram is a tree of propagators and (amputated) 1PI vertices$^1$ [cf. e.g. this Phys.SE post], it is enough to study (amputated) 1PI Feynman diagrams.

E.g. for QED Peskin & Schroeder list 7 possible (amputated) 1PI vertex correlation functions with $D\geq 0$ in Fig 10.2:

This in turn helps organize the systematics of possible counterterms for the renormalization.

Further technical complications come from overlapping divergences and divergences of subgraphs.

For more information, see e.g. this & this related Phys.SE posts.

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$^1$ We allow $n$-vertex correlation functions with any number $n\in\mathbb{N}_0$ of legs; including $n=0$ (vacuum bubbles), $n=1$ (tadpoles), and $n=2$ (self-energies).