So I wanted to offer my thoughts ...
I have kind of a love-hate relationship with Etsy, and I'm sure that's true of anyone who sells there.
What I love about Etsy is that it's easy to use. They charge a listing fee and a percentage of each sale. As a person who isn't selling in anything close to a high volume, it is far cheaper for me to have Etsy be my store than to have a dedicated web site and shopping cart. And even when Etsy's categories fail milliners, there are times when I do get miraculously found.
The trouble is that getting found is hard. Part of that is the category problem; part of it is the sheer volume of items on Etsy, most of which are lousy.
Oh, categories. Emily pointed out how "Accessories > Hat" turns up a huge amount of knit or crocheted beanies. Here's the thing, though: yarn artists have their own dedicated categories! And those categories not only break down to hat, but also the sub-categories of hats (such as "baby" or "beanie"). Milliners have nowhere else to go but "Accessories > Hat." Knitters can file their items under "Knitting > Hat" or "Knitting > Accessories > Hat." Same goes for crochet.
But clicking "report this item to Etsy" is no help. The only options you are allowed are if an item is not handmade, not vintage or not a supply (the three main categories of Etsy) or if it is prohibited or mature content. There's no recourse for category abuse. As an aside, I've seen many sellers write about how Etsy does not do much to police manufactured goods posing as handmade (or vintage). After all, Etsy is still making money off of them.
The problem is that there is so much junk on Etsy. There are some amazingly beautiful finds on Etsy. I have bought some fabulous vintage trims there. And I have admired the work of other milliners and handmade artists who are actually artists. But Etsy adding new categories or fixing SEO* isn't really going to help.
Take away all the crocheted hats, knit hats, surgical scrub caps and vintage hats, and you can still end up with "handmade" hats where the seller bought a manufactured hat base and glued feathers or flowers or ribbon onto it. Add categories for fedoras or any other millinery terms, and you will still have mass-produced hats. You will still have beanies or scarves intentionally mislabeled in order to hijack the results.
Etsy is a great place to host a shop. It's a great way to have an online store without investing in a dedicated web site and shopping cart, so it's perfect for small sellers. It is not, however a place to get discovered. I simply can't expect to ever be found there.
The ideal would be an alternative to Etsy that is juried. No mass-produced goods. No sellers lying about what their item is to gain views. No beanies featuring cartoon characters. (I'm sure that 99% of Etsy sellers do not have a license for their Ninja Turtles or Minion "handmade" items.) That's not to say that knit/crochet should be banned. I would be happy to have my hats appear along side quality knit/crochet.
The good knit/crochet hat makers are getting lost, too, among hats that look like something you stuffed under your bed and forgot about after your grandma gave it to you for Christmas.
*And the search, by the way, is definitely broken. I searched for "felt" in the accessories/hat category ... and some of the results were for felt toys. (Of the two I noted, one was a slug and one was a campfire. Weird. But oddly cute.) I assumed that maybe the sellers used the word "hat" in title or tags to make their items stand out in a category where they did not belong. Nope. Etsy just decided, through whatever bizarre algorithm they use, to return a toy campfire to someone searching for hats.