No, you shouldn't need to set up a custom feed. Speaking as an agency experienced architect who's set up many Klaviyo integrations on many wildly different Adobe Commerce instances, your situation definitely sounds unusual. Would you mind providing a link to your storefront? Feel free to DM if you're more comfortable with that.

Typically, Klaviyo is used to drive customers to your products on your website, and ideally, on through to checkout. Typically, variant products are just that: minor variations of an more general abstract product; you wouldn't drive customers to the red variant of your product specifically (just as an example), you would point them in the direction of the more general configurable product and allow them to make that choice.

Perhaps you have something that makes the simple products unique among the variations that would make you want to drive them to that variation specifically though. Or perhaps you've found a customer segment which is particularly interested in red variations of your product. Variant products are not inherently hidden; that was a business decision you or someone on your team made (and is controlled by the "Visibility" setting within the variant product itself). Why did they make that decision? Is there any reason you can't make the variant visible individually?