"engage", "engorge", "engrave", "engross", "engulf" are all fairly common words that are either often or exclusively pronounced that way (some dictionaries might show /in-g/, but /n/ is really /ŋ/ before g or k, even if they remain). Since these can take prefixes, this also proves we're not limited to being at the start of a word. Searching for words that can be spelled with with "ing" or "eng" finds a few more but nothing super interesting (though a few are in the middle of a word).
Obviously words where "g" is pronounced /dʒ/ (like "j" for those who can't read IPA) aren't subject to this.
You might be right, but for what it’s worth I’ve literally never heard any of those words pronounced that way. I’ve only ever heard the word “English” start with the same sound as inside, while “engage” and your other examples start with the same sound as entertain.
While you're right, I feel like there's no safe argument to make here, because some group somewhere will pronounce some word in a certain way, so there can't really be a blanket rule.
In my local (dialect and) accent all of these words have a pretty clear initial /ɛ/ and not /ɪ/. (But also: /ɪ/ usually contrasts strongly with /i/ here, but the sound before /ŋ/ is almost a third in-between vowel.)
