@Jeroen Vogelpoel , Welcome to Microsoft Q&A!
Thanks for posting a good question.
"Due to this way of working, our billing would still be separated to multiple service providers while we strongly prefer to migrate all our assets, including our domains, to Azure."
-- Thanks for your interest in Azure! Acknowledged, I personally agree with you. But, Azure DNS doesn't currently support the option to buy domain names. To buy domains, you must use a third-party domain name registrar.
The good news is that, the feature to buy domain names is tracked in the Azure backlog. Use the feedback site to register your support for this feature: Azure should be its own domain registrar
As mentioned by our product engineering team in the post "Domain registration is being evaluated and remains on our long term road-map".
At this time, Azure Networking (Azure DNS) it has not been migrated over to the new Q&A platform. We kindly request you to post your Azure Networking/Azure DNS related question on MSDN forum to receive insights and better assistance from the targeted audience. Thanks for your patience & co-operation!