As you well said, CNAMEs are not supported for root domain and also only CNAME records are supported for Traffic Manager: https://learn.microsoft.com/en-us/azure/app-service/configure-domain-traffic-manager.
Due to that limitation this scenario can be challenging, so here is a process that you could use, but the requirements are quite strict.
Requirements
-Your domain should be hosted in an Azure DNS zone to be able to use Azure DNS Alias set (https://learn.microsoft.com/en-us/azure/dns/dns-alias)
-Traffic Manager endpoints cannot be “App Services” they have to be setup as external endpoints
Process
Once you have the custom domain already mapped to the App Services as they are a once off operation, you can proceed with the setup of the Traffic Manager. Create the endpoints , but select the Type as External Endpoint and configure the Custom Header Settings. As FQDN select the Inbound IP of the App Service.
Then if you go to Azure DNS create an A record @.mydomain.com (your root domain) and click on Alias Record Set “Yes” like in the following screenshot, selecting the Traffic Manager Profile. The effect that this will have is that it will update the Public DNS once it is done.