A classic example would be Homer and his wine-dark seas or green honey, to change the descriptions of colour would absolutely be introducing a modern bias but it's a necessary one if you're not going to explain why ancient people had less descriptive terms for colour, a failure to introduce that bias would lead to another unintentional bias in the form of thinking along the lines that ancient people (or Homer in particular) were colour blind.
For a more modern example go look at any works on Project Gutenberg that have recently entered the public domain, it won't take long before you'll find language that at the time of being authored was far less culturally charged or sensitive than it is today, if you were to 'translate' those books an immediate and obvious bias would be to change some of those words to more 'politically correct' ones that would better reflect the meaning and intention of the author, to not do so leads to misunderstandings when people read those works with a modern cultural lens. You can also see this sort of thing with the euphemism treadmill, where older terms can either become taboo or lose all meaning (like mad, insane, mental), and where modern terms can become more appropriate or become taboo (e.g., twitch.tv recently renaming the 'blind play-through' category in an attempt to be more inclusive to blind people).
Modern translations introducing bias and changing meanings like this may seem like a problem if you're inclined to take a more scholarly approach to understanding works by reading multiple translations from multiple eras, or better yet reading annotated works (like what is common in theology circles), but all of this is an incredibly laborious approach to reading that most people, even in universities, don't have the chops for. If you want people to actually read more and read ancient works especially it's better to accept that language and meaning is messy to begin with and to not get hung up on issues with particular translations, because after all there's no guarantee that the person reading the works is going to come to the exact same conclusions, that's something scholars who dedicate their lives to works can rarely do even if overall they converge on a much more consistent understanding of said works.
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I mention this because the video rubbed me the wrong way, the point about how the modern translation 'completely changes the meaning' by using 'human beings' and 'humanity' versus the older translations which use terms like 'men'. It could absolutely be the case that the modern translation is being lazy and substituting mankind/men for the more modern humankind/humanity, where the older translations are specifically using men, as in the male of the species, to better reflect what would originally have been written in ancient Greek. The answer to this isn't something I'm anywhere near qualified to answer (sorry), but what I can say is that men referring specifically to the male of the species rather than mankind is a fairly modern interpretation of the word that's both pushed by people that would consider 'humanity' to be 'hippie cultural marxist bullshit' (as the video puts it) as well the type of people that would prefer 'humanity' as mankind would be seen as a tool of the patriarchy or something along those lines.
If you take the older translations to be more authoritative and correct as the video did you need to understand that to not cloud your judgement with modern definitions and language as the video is doing, humankind only started becoming popular around 40 years ago so if you're taking a translation from Project Gutenberg (at least 75-100 years old) you're simply never going to see such terms and instead it's more likely you'll see 'men' where it's plausible that it could refer to people and not specifically males given the etymology of the word and its historic usage.
A generous interpretation of the video would be that the modern translation is full of such errors and is deserving to be called a 'forgery', but to focus 15 minutes on such a weak example seems so incredibly dishonest and lazy even if the videos interpretation happens to be the correct one. The entire substance of the video is that the older translation is correct because it better suits his particular choice of language, if the translation is so egregious to be called an Orwellian rewriting of history surely there's far more substance that could be packed into 15 minutes.