State theory and specifically the state theory relating to legal practice like "state treaties and processes" is a very small subset of political theory, let alone political philosophy. There is a broad range of political theory outside that narrow subset and "direct democracy" has a specific meaning distinct from representative democracy.
Redefining it as a subset of representative democracy through the introduction of participatory processes makes sense when talking in a context where representative democracies are taken for granted but it is extremely unhelpful when talking in a broader scope where representative democracies are merely one of many categories of systems. There's a reason the PDF references the French Revolution rather than the Paris Commune and describes democracy merely in a (largely ahistorical TBH) European context informed by the Enlightenment era and limits the idea of decentralisation to the federalism of cantons rather than exploring it further: it's not about analysis, it's about creating a national mythology and a historical narrative that has the present system as an inevitable end state that can only be built upon but not fundamentally changed.
And all of this is really a red herring because the original argument was about whether the system Switzerland has makes it unnecessary to appeal to a super-national court in order to force a decision. And given that referendums and public initiatives still require a majority in order to pass the answer is clearly no.
Given that the claim is that the (in)action of the Swiss government accelerates/intensifies the climate catastrophe and that this effectively violates the human rights of a minority group (i.e. seniors are more likely to die from severe weather events like heatwaves) and that referendums have not helped change the (in)action of the government, a referendum is the wrong instrument as the question is not what the majority wants but whether a minority has their fundamental rights violated.
You can argue that the ECHR court was wrong to agree that this is a violation of the rights asserted by the ECHR. You can argue that Switzerland should leave the ECHR and replace it with a different human rights bill - and assuming you have the requisite legal status in Switzerland you can even attempt to force this question with a public initiative. You could also simply argue that they should have sued the Swiss government in a Swiss court of law first (I'm not sure how the Federal Supreme Court works but presumably the process would be a bit more involved) before taking this to an international level. But the ECHR court is ultimately the right addressee when arguing a signatory's violation of the ECHR, "direct" democracy or not.