The way I see it, this is down to two factors:
1. Since Switzerland does have multiple parties and all of the bigger ones a represented in the federal council, there is more compromise and decisions made by consensus. This in turn leads to less dramatics shifts after a term is up and some politicians are replaced. In contrast in the US, where you could have a democratic president and parliament coming in after a term of a republican majority and president (or vice versa). It seems likely, that the new powers to be would set about undoing some decisions made by their predecessors. At the very least you'd have a rather dramatic course change at hand.
2. It is very easy to collect enough signatures for a referendum (50000 signatures within 100 days of the law being passed in parlament) and for every change to the constitution there is a mandatory referendum (no signatures needed, it will brought to a public vote). This means that any controversial law can be brought to a public vote. Which more or less prohibits sudden drastic changes and promotes more incremental change, which brings more stability.
On the other hand, there might be something to the Idea, that direct democracy reduces stability. Basically it is the flip side of 2. above: it is also quite easy to gather 100000 votes to establish a popular initiative. The parties have picked up on that and often introduce controversial initiatives to further their election campaigns. There are not many safeguards in place, so unfortunately there was an increase in badly drafted amendments in recent years. Different parties have brought forward initiatives that contradicted other parts of the constitution.
It has nothing to do with the direct democracy thing, but the most surprising is this so-called "magic formula" is only a tradition and there is nothing in the law that prevents a parliamentary coalition from voting an executive branch representing only 50% of the parliament.
It might be not the only reason for the way the executive branch is set up, but IMO it's an important part of the system as it is.
Even the 'Ndrangheta from Italy has a branch. http://www.thelocal.ch/20160317/mafia-suspects-refuse-to-lea...
There was a politician that forwarded state secrets to Kazakhstan and got bribed. The thing is if you apologize and publicly declare that it was a mistake to get caught, no one bats an eye. She's now head of the parliament. The unofficial rule is, just don't overdo it...
The US population for instance prohibited alcohol and then undid the prohibition, and I think the least that can be said in favor of this process is that since then there's seemingly a consensus in the US wrt prohibition. I'm not sure such consensus is achievable when elected officials enact laws, not to mention when a Supreme Court reinterprets the constitution to de-facto enact a law.
Overall, to me it seems that decisions will be made and reversed no matter what's your process, and it seems that referendums are a great way for the population to "own" the decision - not just affect it but feel responsibility for it.
Also I'd guess that more direct democracy would reduce both stability and corruption (people are more likely to change the status quo, harder to bribe, and not as likely to form 2 or some other small number of groups voting together on every issue as professional politicians.) It seems that we have a lot of both stability (presumably a good thing) and corruption (presumably a bad thing) and it doesn't look like a terrible gamble to attempt to reduce both.