Downside: Sometimes laws can be delayed for 1+ years due to a referendum. The political process is slower and big reforms are much harder.
Upside: Lawmakers need to write balanced laws or they face threats of referendum signature collection from other parties or civil organizations. Often in political discussions you hear that "position X won't stand a chance in a referendum". That is a good thing.
Further additions to your comment. Expanding on your downside: Big reforms like giving women the right to vote only took effect in 1971 on the federal level. On a cantonal level, Appenzell-Innerhoden had to be forced into it in the 90s by the Tribunal Federal, but well.
I'd add some advantages to the upside as well: some changes require a referendum, such as changes to the constitution. But there's more: a popular initiative can be launched and if you collect 100,000 signatures in 18 months, you can force a vote on your own law. This is most commonly done by political parties and adjacent organisations, so it is at least feasible that a privacy-conscious organisation could launch an initiative to make it illegal to store any kind of user-identifying data. It is even possible private citizens could do it. There would likely be a "contre-projet" arguing why this isn't a good idea, but there is often a for/against for any initiative or referendum and they get to present their views in detail (in paper booklets, the vote swiss app, and on the federal chancellery website).
Further upsides: unlike US/British/some other countries, nobody has a 50% voting block in the Swiss parliament and it has remained a coalition since the modern iteration of the country (since 1848).
Basically Swiss politics is extremely deliberative. I honestly think "we will quit Switzerland if they do this!" is a bit of a hyperbolic reaction.
Ah, so the referendum isn’t then scheduled for a date in short order if the requisite signatures are collected but held at the next regularly scheduled election? Fair enough.
Each will have 1-4 issues (approx) scheduled. Elections for politicians happen every 5 years, but no need to wait for those. What takes time (for votations) is the process: you have to verify the signatures once they're handed in at the federal chancellery and then decide when to schedule it.
Upside: Lawmakers need to write balanced laws or they face threats of referendum signature collection from other parties or civil organizations. Often in political discussions you hear that "position X won't stand a chance in a referendum". That is a good thing.