Yeah, I think the original question was a bit weirdly worded which made people focus on web workers rather than complex data in general.
You can use properties (as opposed to attributes) as I demonstrated, and you can use methods like you suggest, but these are both verbose and limited, and add an extra "the component has been created but the props haven't been fully passed" state to the component you're writing. Imagine a component with maybe five different props, all of which are complex objects that need to be passed by property. That's a lot of boilerplate to work with.
I showed earlier how it takes multiple lines and some fiddling with DOM to set a simple property with vanilla web components. Sure, if you're using a framework like lit, you have access to template binding, but at that point you might as well use an equivalent framework like SolidJS or Svelte which just skips the web component layer.
You can use properties (as opposed to attributes) as I demonstrated, and you can use methods like you suggest, but these are both verbose and limited, and add an extra "the component has been created but the props haven't been fully passed" state to the component you're writing. Imagine a component with maybe five different props, all of which are complex objects that need to be passed by property. That's a lot of boilerplate to work with.