Both sentences would do perfectly fine without それこそ. We could say its sole function in these contexts is to add emphasis.
In the first example, the speaker emphasizes how much his friends have come to mean to him. He does so by saying, following それこそ, he cannot even imagine a life without them now. It’s like saying they mean “precisely” that much. What he said after それこそ was meant to back up his claim in a way the listener could relate, because, in general, if one says he cannot imagine a life without something, that thing must be very important to him. It serves the purpose of emphasizing his point well.
The second example can be interpreted the same way. In this interpretation. the narrator emphasizes Mio’s excellent linguistic skills by saying, following それこそ, her level is on par with someone who has lived in Japan for years. She’s “precisely” that good.
However, I can think of another possibility. The sentence explains why Mio was given permission to go out. It implies it’s logical because of her qualifications. それこそ could also be understood as emphasizing the logicalness of the consequence (her obtaining permission) by giving a reference to what is already known to the reader: she is very good at Japanese. She is “precisely” that qualified.
To test this interpretation, I tried changing the position of それこそ as follows:
This eliminates the possibility of the first interpretation because what follows それこそ says nothing but vague ある程度 about the degree to which she has learned manners. For the sentence to sound as natural to me, I would have to already know either she is well mannered or good manners are among the conditions for permission to be granted. Otherwise, what follows それこそ doesn’t serve as a valid reference for the purpose of emphasizing the logicalness of her permission, or anything else.