maybe dropping the background completely on breadcrumbs and going for a thin gray border instead?

I think I suggested that somewhere. Honestly I'd drop both background and borders, to simplify theming.

To complement my previous comment, adding background, border or radii is really, really easy thanks to our utilities. So apart from spacings, I'd say this component should definetely be simpler — and as an argue, the vast majority of breadcrumlbs out on the web are much simpler than our current one.

Let's drop the background, border, and padding then. The main use of a breadcrumb component then is purely to use add a separator between nested items. I can dig that. I'm also thinking we use a local CSS var for the separator: https://codepen.io/emdeoh/pen/KKMOoGx. Note that the first example I tried using a data attribute, but that doesn't work as it doesn't seem to be exposed to the children elements.

Regarding the CSS custom property, we might do as in another place, using an unassigned custom property and our Sass variables as a fallback:

.breadcrumb-item + .breadcrumb-item::before {
  content: var(--bs-separator, #{$breadcrumb-separator)};
}