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I've got a linguistics degree but that tends to mean I would disregard academic literature and focus on how the language is spoken ;)
It's a choice whether to pronounce foreign words 'correctly' when speaking English but the default is definitely to pronounce them in the 'English way' as @Sumo is not alone in finding that it grates...
You're basically right but to be honest (presuming a certain level of Italian ability in Hatch) it's less about his L2 phoneme inventory (as I doubt he really has one) and more about switching from his L1 vowel inventory to his L2 vowel inventory and switching emphasis (and/or downright mimicry). Neither of those meant as a slight, I'm not sure my Italian is good enough to truly have a phoneme inventory and mimicry is an important skill in language learning.
FWIW Italians don't tend to bother with English words because they find the 'oh' sound in English very difficult because their 'o' vowel is so back (i.e. literally pronounced further back in the mouth) than our 'oh' sound which is very much towards the front.
The AMAZING Franco Battiato hit Cuccurucucu is a great example of this when he speaks English during the song (it's also a top tune!):
https://www.youtube.com/watch?v=GuB3f70cYnM
i've been trying to see if there's any academic literature on this, because he uses (most of) the L2 phoneme inventory and phonology when pronouncing non-english names ("borrowed words"?) in the middle of L1 utterances, and as far as i know that's not a common strategy among... er... speakers of language.
^ seem to remember you're a linguistics person but stop me if not