You don't even need to give them actual nics, just veth pairs with routing (some CNI plugons do just that btw).
I think a big part of why overlay nets are so common is that a lot of the engineers deploying these systems have a shaky or non existant understanding of networking and routing, they are stuck in the "local network and a default route" model and if you only have that, overlay it is...
at least that way you can dump them on an entirely different vlan, if you want separation. The best part is: you're using your cloud provider's networking tools (lets be honest it is a cloud and they are way more likely to work)
I know there is simpler ones that don’t use BGP, but I don’t remember their names. For example there is one for AWS that uses ENI interfaces and pass them directly to containers (you’re obviously limited by the number of ENI you can attach to a particular host, then). There is also one that configures routes in the VPC routing table, and a similar one for GCP.