As I understand it, if you don't own the House-elf, then you can't set it free. This means that Hermione's hats would not have freed the elves, as she didn't own them; Hogwarts did. Only the Headteacher, or maybe the staff, could have freed the elves.
So why does Hermione bother trying? Surely she was clever enough to realise that she couldn't possibly free them, so why does she keep trying?
Am I missing something glaringly obvious?
Hermione's efforts betray a general lack of understanding about the role of House-elves as well as her youthful (but largely useless) enthusiasm. Her initial attempts seem to be based on her conversation with Harry about the method he used to trick Malfoy Sr. into freeing Dobby. Note that at the point that she founds S.P.E.W. and begins leaving clothing out for the House-elves, she hasn't even communicated with the Hogwarts elves directly, nor has she been able to find out anything in her 'go-to' text; Hogwarts : a History.
After more conversations with Dobby and Harry (and presumably other elves) her methods become considerably more sophisticated, culminating in her joining the Ministry of Magic where she...
began her post-Hogwarts career at the Department for the Regulation and Control of Magical Creatures where she was instrumental in greatly improving life for House-elves and their ilk.
Because conceivably, the elves belong to Hogwarts as an institution, not to the staff. Including to the students.
Observe that House Elves in hogwarts will give food to students even if it wasn't directed by the staff (James Potter, as per Dumbledore, used the Invisibility Cloak to sneak food. Also, Weasley Twins).
Also, Hogwarts' magical and intelligent property answers to students' needs - the Sword of Gryffindor appears to students (Harry and Neville); the Room of Requirement appears for students.
Because she got no other avenue of recourse for now.
Because JKR decided to drop the Anvil and make a moral/political point, at all costs. Not exactly the most rare move for her, what with "blood purity" motif.
So how do we explain ineffectuous actions? Because Hermione is in some ways a stand in for Rowling. Who is, out of universe, a committed progressive. As such, to her, doing something is more important emotionally than logically assessing whether what is being done is effective. (evidence: Witness the situation with aid to the poor countries in Africa - all the studies agree that most of that aid doesn't get to the actual people who need it and simply enhances the power of the corrupt figures who control its distribution. Or similar things with Ebola. Or giving more money to low-performing schools when all research points to the fact that the money is NOT the reasons, and (lack of) parental engagement is, e.g. as repeatedly documented on Freakonomics).
You mix up intelligence and experience. Hermione has a brilliant mind for sure but is just fourteen. She just thinks that giving freedom to the elves will repair a great injustice (true) and that this is the best thing that could happen to them (completely false).
It is not unusual for a young boy or girl to have very honorable and at the same time terribly impractical causes so there is nothing too surprising in general. Honestly I like the whole story because it is as a nice extension of Hermione's character. Almost all the time we see her behaving as a complete grown-up - prudent and responsible she is the "voice of reason" and usually thinks before acting. With S.P.E.W. she is more like what she really is - a young girl.
Lack of wisdom and zealotry.
While Hermione is very intelligent, she lacks the experience that people or, more generally,
beings have cultures, self-convictions, attitudes, priorities and beliefs leading to behavior which well-meaning people find unacceptable.
House elves are proud that they are "allowed" to serve wizards and witches and it seems that most of them are treated not badly (Dobby, Kreacher and Binky are the exceptions). Dobby had very good reasons to long for freedom, Kreacher was bitter and resentful and Binky was the scapegoat for Crouch.
While house elves are fictional the behavior of humans is identical. Everyone will experience sooner or later that people really sell their birthright for a mess of pottage and you get in trouble for telling them so (and they do it anyway).
If you are confronted with the reality that the being in question does not want your "help" or even find it offensive, you may come to the conclusion that you are completely helpless. And this is a very debilitating feeling.
One option to avoid this is zealotry: Do not try to understand what the real situation is (because it could tear down your ideals), but replace it with a world where everything works as it should work in your mind.
Hermione ignored Dobby's plea that her continued preaching in the kitchen is aggravating and brings trouble for him. This is typical for zealotry: If your intentions are good, the results must be good and if it does not work out, then either the targets of your help will sooner or later convert if you are nagging them enough (wrong) or THEY (You know, those Freemasons/Jews/Pagans/Counterrevolutionaries/etc.) are responsible.
Hermione believes goodwill is enough and therefore her efforts are incompetent. Once she is willing to learn about the house elves and how they think and what they believe she is much better able to transport her message.
