It would be very good to have an API (service method) which would open a given module, refresh dependencies and publish it.

So this would repeat what you can do manually in service studio (open a module, go to dependencies, click "Refresh all", confirm, click "Publish"), or what you can also do using a solution - but for single module. Just to speed up this manual action.

I understand this action takes some time, so such call could start the process and return an ID which you can use to check completion. But normally it takes not more than few minutes, so can be fine even as single request, if this is easier.

Nice to have also a button in service center to do the same. Currently, there are "Redeploy published version" (which does not refresh dependencies), and "Publish all consumers" (which would run a solution for all direct and indirect consumers), but not to refresh a single module.