I ended up settling on Infomaniak's kSuite after looking around. They're a mid-sized registrar and hosting company.
They're partially employee owned (and I believe in the process of becoming fully owned by employees). I'll grant their privacy policy is just standard EU/Swiss boilerplate, though (stuff like no sharing your data, etc., that you always find in EU paid services like this). GDPR compliance was all I was looking for.
The web client looks nice and kDrive is affordably priced if you need a Google docs/photos/drive alternative.
Edits: clarity and me refreshing my memory on their privacy policy