My thinking is that "私" is always in the "universe of discussion".
Yes, and that's why you should always say, for example, 「私は本を買った」, but not 「私が本を買った」in simple sentences, unless "exhaustive listing" is clearly intended.
But in relative clauses modifying a noun, we have to use が or の. は is the "topic marker", and a relative clause does not serve as the topic of the sentence (contrastive wa may be found in some complex relative clauses). We can say either 「私が買った本は面白い」 or 「私の買った本は面白い」, but 「私は買った本... 」 is ungrammatical.
(1) Do native speakers think "私が知っている人は.." sounds strange? Is it grammatically correct?
「私が知っている人は...」 is grammatically correct and sounds perfectly natural.
(2) Would changing "が" to "の" be grammatically correct? Why? Do native speakers think that would sound more natural?
Yes, 「私の知っている人は...」 is also grammatically correct and sounds perfectly natural.
How does the の work in 「日本人の知らない日本語」?
The difference between 私が知っている人 and 私の知っている人 is very subtle, but one may say that the latter sounds a bit more euphemistic and milder, and thus goes well with keigo.