i really like the the "Greatest Marriage". u just need to be open minded in terms of Korean Family System and Korean Law about the family like being a single parent in their country, if not, u will be frustrated to this drama,,hehehhehhe. just have watched only this here..hahha.Goodluck!

The charming stepmom -Taiwanise

Prime minister and I -chinese drama

https://mydramalist.com/31881-princess-sand 

FL is paid by father to look after his daughter to hide from his fiance.

Father then had romantic intentions after his brother proposed marriage to FL to hide the child.  Add in FL first love to mix for more chaos.

Is it me or does this theme become more and more popular these days?

I would recommend 

When the Camellia Blooms - beautiful , beautiful story, full of touching family, friendship and romance moments...  I cried like a million times when watching this from both sadness and happiness 

 Romance is a Bonus Book - the focus in this is more on the job and the relationships than the family.