Part of your mind bogglement is looking at it backwards. The view from a first-world country in the 21st century creates too many barriers, from the existence of formula to the idea that father's have few rights to their children.

Instead, let's look at it from a viewpoint in the far past. Not 1850, but 1350, or 850, or 350 BCE, or 3000 BCE. Let's stick to what I know, Europe and the Fertile Crescent.

All these had a common level of maternal mortality, because nothing much changed. A notable percentage of women died in childbirth or shortly after of complications. They just did. It was the way of the world. So this wasn't a sudden problem they had to scramble to adjust to. They worked out their systems long before there was any historical record, and all these places and times had similar responses to the problem.

The core question is your #3. It presupposes a kind of privacy around the baby, that it must have it's own mother or factory formula to feed on.

When you consider that babies would be breastfed for at least 2 years, think of how many women in a village would be lactating at any time. All of these women were available to feed the new baby. It didn't have to be its own mother.

In fact, there was a now-vanished pink-collar job no man could take over, that of "wetnurse."

Classical Roman authors complain that upper-class women don't nurse their own babies, but act like their nipples are just a kind of beauty spot. In the Odyssey, Odysseus is recognized by his old nurse. In the story of Moses, the princess hires a Hebrew woman who has recently given birth to nurse her found baby. That's how far back it goes that living women hired or purchased wetnurses.

The wetnurses might live in the baby's home, but it was very common to put a baby out to nurse: that is, the baby went to live with the wetnurse in her cottage, as Moses did. There would be considered a natural emotional bond to the nurse, but also her child who was born around the same time, especially with milk-sisters. (In France at least this continued to at least the early 1900s.)

Around 1800, Georgiana, duchess of Devonshire, was considered extraordinary for nursing her own babies.

So with this system in place, you can see that feeding a motherless newborn is not an issue.

Even lacking a wetnurse, the lower classes never flinched at using animals. Medieval orphanages decided the best milk for babies was from a donkey. They didn't use bottles. They just held the baby to a teat. Very handy when you had several foundlings to feed.