No, it is still "myaso". However, rather than being pronounced "MYA-so", the apostrophe separates the "m" and "ya" sounds, allowing the "m" to take up a full syllable. This would make the pronunciation closer to "m-ya-so". I'm fairly certain that this is how it works. You can hear it in this excercise, "Я не п'ю молоко", if you listen closely to the "п'ю".
The letter "я" itself has two sounds packed in it: "й" and "а". Without an apostrophe "й" disappears and instead softens the previous letter. With an apostrophe it keeps the sound. So "м’ясо" is equivalent (sound-wise) to "мйасо", you have to pronounce "й".
They pronounce very clearly here on forvo
I would say "Я не п’ю молоко" when I mean I drink milk in general, but I'm not drinking [this] milk right now or today or at this party. Close to the English "I'm not drinking the milk"
And would say "Я не п’ю молока" when I mean it in general. English "I don't drink milk".
Same would "apply" to Я не їм м’ясо (this meat, meat today, meat at this restaurant etc.) and Я не їм м’яса (I'm vegetarian, I don't like meat, it's not good for my health)
This is not an "official" rule, but just my interpretation and native-speaker feeling...
Yes, I agree that the sentences intended for learners should have been recorded at a slower pace (since no slow audio option is available). Recently a new feature has been implemented where the contributors can record and upload their own audio. However, there is still a bug that I cannot replace old audio with this, only record it where it was missing before. Hopefully, they fix it soon, and then I can re-record this exercise and speak slower :)
I do see how it is easier to learn when it is spoken slower, but that is not how people speak. The Spanish one had a feature to listen to it slow or fast (I'm not sure if it still does or not because I don't use it anymore). I prefer the slightly faster version to train my ears.