I'm not one of the original authors, however I suspect that is because if you are using pipelines-as-code they generally expect folks to clone/copy the original pipeline yaml/json/groovy rather than use an alternate representation via the UI, so it possibly wasn't a priority.
The Admin > Pipelines tab does allow you to download/export to a different pipeline-as-code format, or as you say, you can use the APIs to get the API JSON representation.
For what it's worth you
can fetch artifacts from the result of one templated pipeline into another templated pipeline, but generally you need to parameterise the upstream pipeline's
fetch path from the downstream pipeline template. In other words, when you create an instance of downstream-pipeline from the template and you want to fetch from upstream-pipeline3 you would set the parameter of "upstream-pipeline-name" to "upstream-pipeline3". The fetch task inside the downstream-pipeline template would have the pipeline value set to #{upstream-pipeline-name} and otherwise hard-code the stage/job fetails to match the upstream template.