Hi William - Thanks for sharing your use case here when posting the question.
What you found to be true is unfortunately a limitation with the way our connections currently work. While the relationship between these objects is 1:1, we are not actually enforcing that the connected record can only be used once (this is a valid feature request that we have on our radar).
What we’ve seen work for some users, is to build in some functionality to the live app, via page rules, to prevent your live app users from creating an additional connected record.
While it is not intuitive, you would need to change your connection field to one-to-many (one Client to many ClientCaseInfo’s) and then add a count field to the parent Client object. After you’ve got a count field, you can use that value to hide views using page rules. So for example, if you are looking at a details page of the Client, you would hide the form that inserts a ClientCaseInfo if the count field is greater than 0.
Others may have other ideas. You can look for existing topics such as this for ideas: Enforce Uniqueness/Prevent Duplicates
Hope that helps!