Could something like this happen to Bitcoin?
In other words, are people starting to want to save their BTC coins so much that their value is lost?
I think it already is happening to some extent.
Not that I think its value is already being dwindled down however I do notice, even in myself and a lot others, they tend to save their bitcoins and hold instead of using it to purchase something not that huge anyway. Who would spend a couple of satoshis to buy a water bottle for example? No one. Even if you have 0.0001 sat, you would want to hold on to it and let it grow.
However, it is not all lost because bitcoin is still a currency first and foremost. Despite many opting to hold bitcoin, a lot still use it as a currency to make their transactions. We can see this with businesses using bitcoin to receive payments.
Big companies and some services are using BTC as a means of payment, however, I do think that the trend and the general opinion would stay for BTC to be a big storage of value put into a crypto space with the benefits that it provides for everybody.
As it was said, who said that BTC needs to be a currency for daily life? It's more like a dream for some people.