An Azure service that provides streamlined full-stack web app development.
Unfortuantely I don't think you are going to be able to achieve what you want here with static web apps. Private endpoints are for inbound traffic to the static web app, so by enabling private endpoints you allow resources on your vNet to be able to talk to your static web app, this does not allow for opposite, for your static web app to talk to resources on the vNet.
For app services, as you have done, you would enable vNet integration which supports outbound communication, from the app service to the vNet. Unfortuantely static web apps do not support the vNet integration feature, so all traffic from your static web app is going via the internet, hence why you have to enable public access on your app service.
If you need both to be private, then you are going to need to convert your static web app into an app service site, to be able to use the same vNet integration. Given you already have an app service, it can use the same app service plan, so would not incur additional cost.