I won't respond to all of your interpretations of these few among Tolkien's many awesome female characters, but I will offer a different interpretation of at least Eowyn, who - in my opinion - is the character with the most powerful character arc in The Lord of the Rings.
First of all, where did you infer that Eoryn was 'forced' to care for Theoden? She did remain at his side, out of love and because she knew he needed help, and she was able to give it. In so doing - I would add - she also helped her entire people.
The episode between Theoden and Hama is slightly different from what you presented, but I'd argue that that ditail makes the difference. When Hama points out to Theoden that someone needs to care after the land and the people while he's away, someone of his lineage, Theoden says that Eomer is also riding off with him. And Hama says that he wasn't thinking about Eomer. This suggests that Hama thinks Eowyn is perfectly capable of ruling, just like her uncle and brother.
"Shall I always be left behind when the Riders depart, to mind the house while they win renown, and find food and beds when they return?"
This is, in my opinion, the core of Eowyn's character arc, and one that is frequently interpreted one-way by casual readers. She starts out thinking that only by acting like a man will she prove her valour.
But she gets on a journey, and a lot of things happen on that journey, including seeing her uncle die practically in her arms, and she is unable to do anything to help him.
But something very significant happens to her on the battlefield of Pellenor. She finds herself face to face with the Witch King.
Now, I would argue that Eowyn doesn't find a loophole in the prophecy because there is no loophole in the prophecy. The prophecy states that it doesn't necessarily take a man to do great deeds.
In Middle-earth, everyone is given the opportunity to protect who they love. In fact, to that 'No man am I', I always like to add, 'Thank goodness!'
In Tolkien's world, not just powerful male warriors can change the world. Eowyn and Merry proved it on the Pellenor Fields that sometimes a male warrior is actually not what it takes. And that's what Eowyn also learns.
I often hear people lamenting about Eowyn finally marrying Faramir. And I wonder, is it a sign of weakness to love someone and want to share their life with them forever, create a family? What is it that disturbs some people so much about it? Is a woman supposed to remain alone forever in order to prove to be independent and even powerful, whereas a man's dedication to his family is considered praiseworthy?
When Eomer doesn't find Eowyn alongside his uncle, thinking she also fell in the battle, he protests against it because she deserves the same honours his uncle is receiving. And when he discovers she is alive in the Houses of Healing, he drops everything to be at her side for days until she wakes - and he's the king of his people at that point.
I won't enter in all the details of Eowyn's incredible character arc, but I wanted to offer an alternative interpretation to yours because it is possible to read Eowyn (as well as the other characters you mentioned) in a very different way.