Replacement for "Copy Parent & Child Records" Button Creator?

Yes, the buttons still work. In fact, you don't have to use the generator. You could copy and paste the Javascript into a formula URL field yourself and it will work. It seems to be whitelisted. Hope they don't change that.

Maybe another way to do it could be through the API and a formula button, but that would be a hassle if you can just do the above.

Yes, the native button is definitely whitelisted for the indefinite future so you can trust it. It also has the advantage of the user interface that it makes it easy to land the user on the newly created Parent record.

There is an API method, but it's a little more awkward to use because you don't have an easy way to land the user on the record. What I tend to do in those situations is to create a report filtered by records created by the current user in the last minute and then land the user on that report which will just have one record on it.  

My preferred method is do this with a Pipeline.  It provides for finer control over the process.  I also document somethings in the process for future reference.