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?
This could happen to Bitcoin, and I believe we need to distinguish between bad money and good money.
For me, Bitcoin is the good money, while fiat is the bad money. Why? As stated in "Gresham's Law," bad currency tends to drive good currency out of the market. When we refer to the market, it could mean investment, which means Bitcoin gets pushed out so people can use it primarily as a store of value, while fiat is used for daily transactions.
However, this doesn’t mean Bitcoin will become obsolete or lose its value. Just like gold, it’s not used for daily transactions, but its value appreciates over time.
I can liken the concept to your narration, Bitcoin is the good money because it's a store of value and that is why people hodl it. Fiat becomes the bad money that is commonized and people prefer to use it for the payment of their daily needs. Most people that holds Bitcoin will rather not spend it because they know that the value will continue to increase in the future, so they'll rather spend fiat that can be vulnerable to inflation. Although in practical the high value of Bitcoin is making it to deviate as a digital cash that it was created to be. I however don't believe that the value of Bitcoin will be lost because people will rather hodl it than use it for p2p transactions. It can be likened to a valuable asset like gold, despite not using it for payment, it's value still retains.