I largely agree with your first paragraph, but I do think the term “birthing parent” is important in contexts such as parental leave policies.

“Mother” could be inadequately specific in situations involving same-sex relationships, surrogacies, and/or adoption. For example, Washington state offers up to 6 weeks of medical leave to the birthing parent, in addition to the 12 weeks of bonding leave that are offered to both parents.

Policy language needs to be specific and able to accommodate minority or edge cases.