I’m not generally inclined to be too critical of other people’s pronunciation of ancient Greek, since it’s a dead language, and there have been many different historical periods and regional dialects, each with its own ways of pronouncing things. Generally all that matters is that you pick some reasonably standard system of pronunciation, from among the ones that are in current use. There are three popular choices that I’m aware of: (1) Erasmian (which comes in multiple flavors), (2) reconstructed Attic, and (3) modern (which is what native speakers generally do, often based on a philosophical and aesthetic feeling that they want to emphasize the continuity of their culture). You can also pick whether to use stress accents or tonal accents.
It doesn’t really matter a heck of a lot which of these you pick. Just pick one and train your brain to use that consistently.
Having said that, I have to say that the pronunciation on the linked web site seems pretty poor to me. He basically seems to be attempting an Erasmian pronunciation with stress accents, but he’s not consistent at all. In some cases, he’s clearly been influenced by modern Greek pronunciation, which is fine, but he doesn’t do it consistently. He pronounces γ the ancient way in most cases (hard “g” as in “good”), but in one spot he pronounces it the modern way (γε as “ye”). During the English-language introduction, he pronounces Εὐθύφρων as “Efthifro,” which is a modern pronunciation (ευ=ef, υ=i, omitting the final ν). Using modern pronunciations is perfectly fine as a choice, but randomly mixing them in here and there seems bad to me.
His pronunciation of the vowels seems inconsistent. He pronounces Σώκρατες as “Socratis.” This pronunciation of ε is not standard in any dialect I’m aware of, and it isn’t how he pronounces it elsewhere. Probably he’s just slipped into pronouncing the name the Anglicized way. He pronounces ἐνθάδε as “enTHAday,” with a different vowel for the initial and final epsilons. As far as I know, both of these ε’s should be pronounced the same. The pronunciation of ε does differ between different pronunciation systems, but it’s generally always the same within a particular pronunciation system.
He pronounces υ sometimes like English “flute” and sometimes like “impute.” Again, it would be better to pick something and be consistent. Different people who use the Erasmian system do pronounce this vowel in different ways, the most popular being like French “tu,” although I do “book” because I have trouble with the French vowel. If he wants to use “flute,” that may actually be historically correct for the Homeric period, but probably not for Plato’s dialect. Pedagogically, it may be a problem because then it’s going to sound the same as the diphthong ου, as in words like νοῦσον. Anyway, if he’s going to use that, he should use it consistently.
A couple of more minor issues that are not necessarily mistakes: –
To my ear, he seems to pronounce omega the same as omicron, both being like English “scope.” Plenty of people who use an Erasmian system do this, but it does erase the distinction between the two vowels, so that in your brain’s memory banks certain words will become homophones.
He pronounces the ει in διατρίβεις like English “hay.” I think the historical reconstruction is that in most of the Homeric and classical period, in most regions, ει was not diphthongized, but some users of Erasmian do pronounce it like “hay,” e.g., that’s the way it’s presented in Pharr.
For comparison, here is my attempt at pronouncing the first sentence, using Erasmian with tonal accents: https://soundcloud.com/benjamin_crowell-tutanota-com/plato3 . I think you can mostly hear me doing the tonal accents correctly, although I sometimes slip into stress accents, because my native language isn’t tonal. I try to pronounce omega and omicron differently, which I think is audible in the word νεώτερον, although I think I didn’t always succeed in other words in the sentence. (That vowel distinction isn’t part of my native Californian dialect, so I have trouble producing it consistently.) I do ει as a diphthong of ε and ι, which is probably ahistorical but helps me to get my vocabulary organized in my brain so that I know from its sound that a word like λείπω has “ει” in it and not some other vowel or diphthong. I pronounce η like “baby” and ε like “hen.” I’m not saying that mine are the only correct choices, but I do try to get my brain programmed to do my choices in a consistent way.