General Architecture confusion

I'm a bit lost, or perhaps a bit surprised. I have Zen-Cart up and running as a test to get a sense for it and it seems the idea is that Zen-Cart is your website. And that you would customize the Zen-cart template to get the site you want. It seems to me that this is backwards. I would expect Zen-Cart to integrate into a website rather than being the website.

For example, lets say I have created some great theme on Wordpress or some other platform or just from scratch or from a purchased template. Now I want to add a shopping cart. Ideally it seems I would want to install a cart that has snippets of code or widgets that are inserted into my existing website. The snippets would be products, shopping cart indicators, shopping cart view, checkout, etc. that I would strategically insert into my existing site's theme pages.

Perhaps I can easily do this with Zen-Cart, but I have yet to run across documentation that describes this. Please point me in the right direction if so.

It's a matter of perspective.
Evidently you think that because you're already using WordPress that WordPress should be your "website", and everything else should just plug into it.

Unfortunately, Zen Cart is more than merely a "shopping cart". It's an advanced eCommerce system, intended to manage your entire catalog, your merchandising, your promotions, your inventory, as well as the actual shopping cart and checkout part that you're trying to "bolt on" to your WP site.

So, yes, Zen Cart is built to be your website. Or at least a section of your website. If you want to stuff it into a subfolder of your site, feel free. But it's not intended to be hacked apart into pieces running on another system. There's a lot more going on inside than that.