I haven't watched since early season 3, but I was curious to know what people thought of it now. But based on my prior viewings, I agree with what you're saying, especially about them trying too hard to make the characters contemporary and "relatable," which I find applies to both Discovery and SNW. While I can imagine some people disliked the formality of speech that previous Trek series employed, I liked that people spoke in a more "high class" way, for lack of a better term. It spoke to the kind of future society that Trek was representing, and made those series feel more timeless to me due to their lack of colloquial speech.
Side note: I did find it annoying when they would have a character be too into some kind of 20th century thing - Tom Paris being the worst, Archer's inexplicable love of water polo, etc.
Having characters act too "chummy" and casual, using current slang terms seems out of place in the more formal military-type hierarchy and semi-idyllic society of Star Trek. Also, you can have the occasional character who acts more informally, but when too many characters share those traits, it's too much. For example, Dr. McCoy was often informal and spoke his mind, but he was pretty much the only TOS character other than Kirk to act that way, and it made sense for both their characters. Plus, they were both high ranking officers on the ship, so it makes more sense that they'd be allowed some more liberties with how they spoke and interacted with other crewmembers.