How did they know about 3 years they weren't going to have a child? what if they had? What would have happened to Frank??

This echoes Austen's own brother Edward's adoption by wealthy relatives, he took their name when they died and he inheirited.

There may be echoes of Fanny Price too, she's "adopted" by the Bertram's in Mansfield Park.

This indicates that the practise was likely widespread

This was common at the time - someone had to step up to play mother / become the lady of the house. Fanny Knight (Austen's niece) did it. This would be considered parentification now. (Definition "the assumption of a parentlike (or adult) role by a child.")