"Assign Permissions to Contents" need not be used everytime user is added to a group.
You need to first verify if the corresponding projects/workbooks are setup with permissions for the appropriate groups.
If your tableau server is using AD authentication then you would need to sync the group on the server. If you are on Local authentication, then it is just sufficient that the user is added to the group provided the permissions are setup properly.
Generally when a new workbook is published to tableau server the project permissions are defaulted to the workbook unless the publisher modifies it at the time of publishing on the publishing window of tableau desktop (or) after publishing, then permissions are explicitly modified. If the permissions are deleted while publishing then "Assign Permissions to Content" option needs to be used to cascade the project level permissions to the workbooks/view published to it.
I absolutely agree that it should work that way. That being said we have tested that and found that when I add a user to a group they do NOT inherent the group's permission. What would be set incorrectly to make that happen? Remember that I am adding a user to a group after that group has already been given permission to interact with the project.
check out the below KB if you may not have already; just in case the permissions are setup as 'Deny' at the project level.
If they are setup properly and yet the permissions are not being inherited then it sounds like a bug behavior to be reported to tableau support unless we missed something.
Has the individual user been denied permission to the project? Are they part of a role denied permission to the project? Is their default group denied permission to the project?