Yes, you do have to dig a bit. Hulu tends to have the older network shows these days, and Amazon has older PBS and HBO shows, but Netflix still has some good stuff:
If you like well acted non-American detective/thriller shows, Broadchurch, Happy Valley, Bodyguard (more thriller), and Secret City (Australian thriller) scratch that itch. Or for a Western version, Longmire. Miss Fisher's Murder Mysteries is much lighter than the others, but the period setting (and costumes) are great.
One Day at a Time (remake with Rita Moreno) is a terrific light comedy. Kim's Convenience is also very charming, in a similar vein. Parks and Recreation is similarly warm, although its subject is the workplace rather than family.
Episodes is a darker comedy about two British TV writers who move to Hollywood to work on a remake of their British hit. Matt LeBlanc (Joey from Friends) plays a fictionalized version of himself, which is much funnier than you might expect.
They have all the Star Trek series if that's something you'd enjoy.
Sherlock (the BBC version with Benedict Cumberbatch and Martin Freeman) was a lot of fun in its early episodes.
Gilmore Girls, a dramedy about a single mom and her teenage daughter, is feeling a bit dated these days but I still enjoyed it and the dialogue is terrific. Speaking of teens, they also have Glee (the teen musical comedy about Glee Club) which was pretty groundbreaking in its early seasons.
Mad Men is just terrific, and so was much of the West Wing, if you want television classics.
No Tomorrow was a quirky little dramedy that ran one season on the CW. It's about a guy who becomes convinced that the world is ending and decides to live his life with no regrets, and talks his (sort-of) girlfriend into doing the same.
If you like family dramas like This Is Us, Parenthood was a similar series. The writing was not quite as consistent, but the good plotlines were really good and the cast involved some talented actors.
The Netflix Marvel series (Daredevil, Jessica Jones, Luke Cage, Iron Fist, The Defenders and The Punisher) have their ups and downs but have some interesting moments--season 1 of Jessica Jones is my favorite. There's also the CW family of superhero shows--I quite liked the first couple seasons of the Flash, and I have heard good things about Legends of Tomorrow (the first season was so so but apparently it's improved)--they also have Arrow, Supergirl, and Black Lightning.
For science fiction, I really liked the remake of Lost in Space, and I thought Colony (LA after an alien invasion) was solid. iZombie is one of my favorite shows--the plot is completely ridiculous but the gist is that the main character takes on the personalities and some of the memories of the people whose brains she eats, and she uses this information to help the police solve their murders. The real highlight is watching the very talented Rose McIver play everything from an aging rock musician to a cheery preschool teacher to a flighty teenage girl. I also enjoyed Lucifer and Stranger Things.
The Good Place is one of my favorite current series--it's a little hard to explain, and you don't want to know too much going in--but the gist is that it's a comedy about the afterlife starring Kristin Bell and Ted Danson.