It's not only about the better pricing, I very much enjoy not buying cheap knockoffs from Amazon.

You want to buy product X? How about you get this genuine "some name that is clearly made up on the spot to look like a brands name"? It might give you cancer, but that's a low price for a very low price, isn't it?

This is basically what Zalando and Otto are.

I guess I want a search engine that searches all the European online shops for the products I’m looking for, lets me browse and compare

Yes. This so much. I don't want to search 100 different shops. I don't want 100 different accounts, different accepted payment methods, different delivery methods. Something like Amazon, but not controlled by one huge corporation, would be so great.

The closest we have are price comparison sites, but once they are big enough they don't list merchants that don't pay them for that listing, so you often don't find the lowest price there either.

I also want the companies to start selling their products through European online shops, instead of just Amazon.

Well in most cases, manufacturers don't choose the vendors but the other way around. I'm sure most companies would love to be available at every single merchant existing, including European ones.

If you don't need an item within a limited time frame, you should ask European sellers if they can get the item and sell it to you. Might not always be successful, but at the very least it helps them to know the customers' demands better. European merchants aren't Amazon-sized and can just store and offer every existing product so they need to know what we want.