Could they have survived? A few prominent multi-ethnic states in Europe survived after all. What would it take?
I think the issue is that, with the exception of the Romansh, Switzerland didn't really contain any ethnic groups that didn't have a nation-state outside of Swiss borders. When the Sonderbund Compromise of 1847 failed after only five months, the resulting Swiss Civil War attracted major influx from Austria-Hungary, France, and to a lesser extent Piedmont-Sardinia, until Switzerland, alongside Italy, became a center of the Austro-French proxy conflicts of the mid-19th century. Whilst the German Swiss retained a historical animosity towards Austria and a nervousness towards autocratic Prussia and the later German Empire, the French and Italian Swiss, as a result of these conflicts, drew closer to their respective nation states in ideology. After the Entente victory in the Great War of 1914-19, the French and Italians annexed their respective minorities from Switzerland, which was the beginning of the end. After German authoritarianism was unmade by the 'Red November' unrest that established a stable democratic-socialist republic, the rump Swiss states' vote for union was all but inevitable after the "Munich Miracle" that was the Liebknecht economy resulted in, against the odds, a Germany enjoying a standard of living higher even than the French.
As for the other multi-ethnic states of Europe, I can only think of two that survived in any form: Belgium, under well over a hundred years of careful British supervision, and the Danuvian Federation, which was established by Bela Kun with German support as a buffer against Russian expansion in the Balkans (which after the 'Black Barons' became the power behind the Tsar's throne was a genuine horror) and still involved its fair share of coercion. The Crown Union of Serbia and Croatia lasted a while as well, but of course the circumstances of its dissolution do not make it a model of stability.