Always chop forests on mines, always chop before placing a district on a forested tile. For forests on flatland, it depends. Definitely chop for the key wonders like Pyramids, Oracle and Kilwa and chop for important infrastructure but if the city is really production starved, leaving some forests for lumber mills can be fine. Lumber mills just unlock a bit too late for my liking.
Chopping for settlers is very useful. It's perfectly fine to gimp one city a little to churn out more cities faster. Magnus chopping settlers is such a huge production boost for your game, the earlier you place new cities, the earlier you place down their districts and the cheaper those districts get. But I will usually wait to start the chopping till after Colonization is plugged in and the Ancestral Hall is built to maximize the efficiency of the chops. I usually try to build 2-3 settlers before this, and might chop some of them, but prefer not to.
Also, chopping out Horses for a Horseman rush is one of the strongest moves you can make if you have an attackable neighbor. Again, gimping some cities to gain more land usually ends up worth it in the long run. In civ 6, more cities generally beats out better but fewer cities. Legion chopping is also a classic strategy as Rome, legions chop out more legions and snowball to kill your neighbor.
I think the main flaw in your idea is that early production snowballs really hard in this game. Lumber mills come late, having a really strong city at that point in the game is not quite as useful as the early production boost of chopping. Later in the game, you can easily create 3-4 powerhouse cities by centralizing trade routes. Internal trade routes pre-tier 3 government will provide decent growth and production and if you have a commercial/harbor in each city (which you usually should), you can have 14-20 trade routes easily. With democracy or communism, each trade route can be 8+ production, so most of my late game production comes from trade routes, if I'm not playing a civ with good industrial zone bonuses or a strong coastal civ with lots of coastal cities.
But for the most part, I do avoid chopping flatland forest unless absolutely necessary. Forests on hills however, should absolutely be chopped every time. Mines are just better, and you can get both the instant production boost and a strong productive tile. Saving some chops can also have some value by the way, you don't need to chop immediately, and chops scale in value with tech and/or civic progress. Chopping in the final steps of a science victory is very common (buy spaceport, chop in projects).
One other thing you haven't mentioned is food chops. Chopping food resources, while unrelated to this forest issue, is critical in getting newer cities up to speed, and to get cities to pop 10 for rationalism.