Swiss medievalist here. At last a question I am qualified to answer on AskHistorians.

There still is Swiss nobility. It's just very rare. E.g. the family of the current major of the Swiss capital Bern, Alec von Graffenried, appeared in the sources in 1272.

So why is it so rare? It is so rare, because Switzerland is deeply republican. While the Swiss democracy probably did not really start in 1291 as they want you to believe, republicanism or at least a strong republican tendency, is a bad environment for nobility. Nobility is always under pressure, families die out - and they are being replaced. And it's this replacement that did not happen in Switzerland to the same extent as in other countries.

Roger Sabloniers book, Adel im Wandel (Nobility in Transition), does a very good job describing the economic problems of the noble families around 1300 in what is today Northeastern Switzerland - but was not back then. It was Habsburg territory. The Habsburg went East and took a lot of noble families (talented lower nobility!) with them. Higher nobility would not move, but came under significant economic pressure with the rise of the cities and the economic power of the merchants. Monasteries expanding in the 13th and 14th century. It was tough and less and less noble influence and protection in a more and more republican environment. For centuries.

A very significant event was the battle of Sempach in 1386. It was basically the Swiss against the Habsburg nobility. The Swiss won. And the nobles died. (Look up the very good article in the Historical Dictionary of Switzerland https://hls-dhs-dss.ch/de/articles/008871/2012-12-18/ - deepl.com is your friend if you do not read German). There are families who lost 5, 7 males in that battle. I remember transcribing passages from a manuscript from the monastery of Engelberg with the names of the dead. It was devastating. And these families never recovered - in the now more and more republican environment.

In 1415, then the Swiss (namely Bern!) invaded Aargau, the Habsburg heartland, there was not very much nobility left. A notable exception were the nobles of Hallwyl. Two brethren. One in Hallwyl, one on Wildegg half a day away. Hallwyl was handed over, Wildegg was ignored next to the road the army used (why!?). After the war, one of the brothers joined Bern, became a citizen and his grandson knight Hans von Hallwyl played a significant role in the battle of Morat in 1476. The other brother left Aargau in 1415 and went to Alsace and played an administrative role in the Habsburg government. This is an interesting example of how a well-off family coped with the pressure. It was agile enough to join with the new "lords" or find a role in a neighbouring territory. Other nobility in Aargau was washed away and the Swiss certainly did not allow any new nobility to form in Aargau in the next 400 years.

In fact Bern was a good place for nobility, since the more and more powerful city the town republic with biggest territory North of the Alps!) was in need of talented nobility for administration and to govern the vast territory. So perhaps a reason while there is still a bit of nobility in Bern. And also in some of the other Swiss cities like Zurich or Basel.

Sources:
- Historical Dictionary of Switzerland
- Roger Sabloniers: Adel im Wandel, Zurich 2000.
- Carl Brun: Geschichte der Herren von Hallwyl, Bern 2006.