Yes the dairy cycle is more complex I simplified it because I wanted to focus the impregnation part.

Milk production decreases greatly after the first 9-12 months. To make it profitable they do get impregnated repeatedly. The life of a dairy cow typically goes like this:

• 15 months old: First pregnancy
• 24 months old: First calf is born
• For around 12 months: milking
• 60 days "dry off"

The cycle is then repeated. Since pregnancy and milk production is taxing on the body and milk production declines, most cows get slaughtered at 5 years old with an average of 2.5 pregancies (average lifespan is 20 years). This also makes sense because to maintain the herd you need to keep the number of females stable, which have a 50% chance of being born (male claves get slaughtered ofc).

Maybe some homesteads or subsistence farms keep milking them for years after one pregnancy, but otherwise even for free range grass fed whatever, if they sell milk to make a profit this is how it goes.

You can get all this info from industry sources.

These practices are not particular to factory farms. Some farms do keep bulls but this is costly and impractical on the long run for genetic diversity reasons, so most non-factory farms also buy sperm and do the artificial insemination.

Also "local" as an adjective doesn't mean much in terms of practices. All farms are local to somewhere. In a 50km radius of where I live there have been investigations in at least 10 farms that found severe animal abuse and neglect in the last 5 years. Those were all local farms that got to put a nice local stamp in their products.