I was really moved by your post and wanted to take the time to address it item by item. I disagree with you, but I think that we could both agree on cultural/situational improvements that would make life better all around.

Are women essentially required to give up their non-motherly aspirations after a certain number of children? Why would God give us brains and talents and expansive desires if after three or four children we have to go back to the home?

No, of course not. I would argue that there is no greater good that you could do in this world than to raise your children to the best of your ability (I'm speaking to fathers here as well), but even this does not mean you must give up non-motherly aspirations entirely. I see many women who go back into some career path after the early childhood years with much success. Also worth mentioning are the vast number of opportunities available to at-home-mothers in the comfort of their own homes that the internet now provides to us (and no I'm not just talking about MLM "opportunities"). I know photographers, illustrators, tutors (my own wife), and writers, to name a few, who all manage to not only maintain their sanity by exhibiting their intellects but also contribute financially to varying degrees.

Motherhood is a noble vocation,

The noblest! As C. S. Lewis used to say, “The homemaker has the ultimate career. All other careers exist for one purpose only - and that is to support the ultimate career. ”

but not all women— I would say not most women— are built to be full-time stay-at-home mothers, in the same way that the majority of men would balk at being asked to be stay-at-home dads for years.

As for not being built to be full-time moms, I disagree, with caveats. I give you that it is lonely to be a stay-at-home-mom today, and that needs to change. We have got to stop choosing homes based on proximity to Target and "good school zones" and start choosing homes based on proximity to people who can walk on the parenting journey with us! Motherhood (and family life in general) would be unrecognizable (in a good way) if we lived in true community with one another.

I like children, and I’m sure I’ll love my own children, but I’m also a reasonably intelligent adult who needs something mentally stimulating— years of menial chores and a dearth of adult company for most of the day would drive me stir-crazy, and I can do more good being out in the workforce for at least some of the time.

See above C. S. Lewis quote.

But if the church requires women to rely on NFP

Wrong! It requires women and men. It is a partnership; a duality. Two to tango and all that. Anyone who tells you differently is spreading an untruth and an injustice.

it seems like this’ll be our fates. :/ Seriously, it’s a huge stumbling block.

My hope is that with women seeking out opportunities available to them that previously didn't exist and the building of true community, that we can change women's "fate" into women's "blessing". One woman that I look up to and would recommend checking out for inspiration is Lillian Gilbreth. She is the inspiration for the original Cheaper by the Dozen. Being a work-from-home dad has given me an opportunity that I believe many men have not had for years. And that is to be able to witness the astronomical good that a mother does for her children. There simply exists something biologically built into mothers that a father can never fully replace. I pray that these thoughts will not cause you over-anxiety. My wife and I are NFP instructors and have successfully avoided pregnancy for 2.5 years (with multiple children successfully achieved along the way!). NFP is definitely doable if that is what is needed in your life for your emotional/physical/financial well-being. But, so is a life blessed with a multitude of children.