That site has a review of 22 audio files, of which at least a couple are in reconstructed Koine. They’re what you’re after, rather than Erasmian or Modern Greek.
That said, read the answers to:
- Is the modern pronunciation of Greek accurate for koine?
- What are the pros and cons of the Erasmian pronunciation?
The Koine of the 1st century AD was a system in flux—and most of the flux had already ended up where Modern Greek is now. I buried in comments to my answers there this:
Εν τω κόσμω æν και ο κόσμος δι’ αυτού εγένετο και ο κόσμος αυτόν ουκ έγνω. Εις τα ίδια ǽλθεν και y ίδιy αυτόν ου παρέλαβον. Όσy δε έλαβον αυτόν έδωκεν αυτýς εξουσίαν τέκνα θεού γενέσθαι τyς πιστεύουσιν εις το όνομα αυτού. y ουκ εξ αιμάτων ουδέ εκ θελǽματος σαρκός ουδέ εκ θελǽματος ανδρός αλλ’ εκ θεού εγεννǽθæσαν. Και ο λόγος σαρξ εγένετο και εσκǽνωσεν εν æμίν και εθεασάμεθα την δόξαν αυτού δόξαν ως μονογενούς παρά πατρός πλǽρæς χάριτος και αληθείας
Apart from the etas and upsilons, and the bilabial pronunciation of phi and beta, it pretty much already was Modern Greek. And conversely, if you’re going to teach to spelling, Erasmian (or for that matter reconstructed Ancient Greek) are a more sensible choice anyway, as a stable target lining up with the spelling. It is in fact what you’re asking for, OP (“both the aspirated consonants and the diphthongs right”), as opposed to what was likely spoken at the time.
If the purpose is teaching, rather than jumping in a time machine to hear Greek spoken the way Mark and Paul spoke it, I’d stick with Erasmian.