thanks for the question. I can see that in 3.4 it was possible to sneak in some content via that hook “Template::Layout::Backend::HeaderActions“, but since it was moved to the vue.js component we did not consider good alternative for that use case.
So in short term most control you would have if you directly edit vue.js component TopNavActions.vue . Than you need to rebuild the ui, in the ojs folder with commands `npm install && npm run build`.
For longer term I created github issue, where I appreciate if you provide more details about your use case so I can make sure its well supported when we improve extensibility of this component.