Plan was simply to contact and show them some art samples and ultimately the goal to be included in their products. Is there a better way to go about this?
A better way to go about this? Not necessarily. You can add some bells and whistles but that's essentially the sales process for anything in a nutshell.
A better place to target? Maybe. I'm not super familiar with the art world but a quick search yielded pages upon pages of brokerages focused on selling other people's works.
Etsy strikes me as a 'sell your own stuff direct' model so even the best sales process is swimming upstream. Someone focused on other people's products (essentially a retailer) could be a much better fit. Your incentives (get your art listed) and their incentives (find other people's art to list) are aligned.
Proper targeting is the highest point of leverage in any sales/marketing effort. Let me know if you need a hand and we can work out options in some more depth!
I have worked on branding for a variety of companies. I love helping artists share their work with the world.
Contacting sellers on Esty is a great idea IF your designs add value to their products. A good way to find out is to create and show them some samples of how your design would look on their product.
If you want to hear some more ideas, or ask more questions- please set up a time to speak.