Why does the Greek “αγγε” transliterate to “ange” and not “agge” in English?

Ah, a Modern Greek perspective in the question details.

I answered the corresponding Ancient Greek question at Nick Nicholas’ answer to Why has the word συγγεής two γ? I know it comes from σύν + γεν, and that later the ν disappeared, but why putting two γ? And why has the ν disappeared at the certain point in history?

Tl;dr:

• The velar nasal /ŋ/ is not identical to the alveolar/dental nasal /n/. It is, if you like, a cross between a /ɡ/ and an /n/.
• English, Latin, and many other languages have chosen to write it as an <n> before a <g>.
• Ancient Greek instead chose to write it as a <g> before a a velar: <gk, gch, gg>.
• Which is not absurd, given that the sound is a cross between a /ɡ/ and an /n/.
• Ancient Greek had geminated voiceless stops; /kk/, /pp/, /tt/. It did not have geminated voiced stops; any words with /dd/ or /bb/ are not native Greek words. So there was never any risk of orthographical <gg> being interpreted as [ɡɡ] instead of [ŋɡ].
• Modern Greek uses Ancient Greek historical orthography, so it was not going to respell the sound if it survived unchanged from antiquity. It still spells it as a <g> in front of a velar.
• So yes, it would indeed be strange to write Angelopoulos as <Angelopoulos>, Ανγελόπουλος. Standard Greek Orthography never has (though I think Soviet Greek orthography, which was phonetic, did).

In addition:

• Modern Greek pronounces <gk> and <gg> (historical [ŋk], [ŋɡ]) identically as [ŋɡ]. Just as it universally pronounces <nt> as [nd] and <mp> as [mb].
• Many Modern Greek dialects, and increasingly Standard Modern Greek, pronounce <gk>, <gg> (historical [ŋk], [ŋɡ]) as just [ɡ], dropping the initial nasal—just as they have done with orthographical <nt>, <mp>. So the usual pronunciation of the name Angelopoulos is actually [aɟelopulos] anyway (palatalised g). One more reason why it would not occur to anyone to write it with an <n>.