How do I fathom the 3rd declension?

And I weigh in too, though my answer is not really different to Desmond’s.

The way to fathom the 3rd declension is via proto-Greek. That’s what the grammars do, whether it’s the most useful thing to do or not.

Focus on the recurring endings: -(ς) -ος -ι -α -Ø, -ες -ων -σι -ας -ες (or -α -ων -σι -α -α in the neuter).

Focus on the genitive singular, because that always clues you in to how the noun’s declension actually works. ἅρπαγ-ος > *ἅρπαγ-ς > ἅρπαξ.

Don’t focus on each distinct pattern of the third declension. For passive recognition, it isn’t worth it.

Resign yourself to the fact that the nominative singular will be strange.

Treat the dirty vowel stems (3, 4, 5 in your list) as separate declensions. You can, with enough concentration, discern the recurring endings there too; but you might as well not; there’s been too much intervening sound change to make it worthwhile.

Do not even bother learning the highly irregular patterns, like ναῦς and ἀνήρ and γυνή. They’ll be frequent enough that you’ll pick them up from context anyway.