Embedded Service deployments can be tricky because of the number of separate components that make them up.
Before you begin, you should ensure that your Embedded service config has a short name - 3 characters or less is ideal, like Web or Bot - and the Developer name of the channel should be less than 21 characters. Any longer, and your deployment may fail with the following error:
{"message":"Invalid EmbeddedServiceConfig DeveloperName."} or force-app\main\default\networks\<name of your network file> Label: data value too large: <extended network name generated by salesforce> (max length=80)
Once you've double-checked the length of your names, you're ready to deploy. The full list of metadata types required for a successful Embedded Service Deployment is as follows.
Branding setCustom siteDigital experience bundleDigital experience configEmbedded service configMessaging channelNetwork (Community)Static resource
Once you have added those metadata types to your filter, and selected your changes, you should be able to deploy them all at once.