while it may be true that, just because, nevertheless, not necessarily
Sentence 1.
This coordinate conjunction is used to connect two sentences. It acknowledges that the first sentence is true, but warns that the expected or logical conclusion from that fact does not necessarily follow. It is often used to introduce a counter-argument or a limitation to a general assumption.
The phrase starts the second sentence to contrast the premise of the first sentence. It functions as a transition meaning 'even if that is the case' or 'just because of that'.
彼はお金持ちだ。
He is rich. Nevertheless, it doesn't mean he is happy.
安いレストランだ。
It is a cheap restaurant. However, it doesn't mean the food is bad.
スポーツが好きだ。
I like sports. But that doesn't mean I want to watch every game.
This expression is frequently paired with negative endings like ～わけではない (it doesn't mean that) or ～とは限らない (it's not always the case) in the second sentence to emphasize that the conclusion is not guaranteed.