As I said, bitcoin is unlikely to become a bank in the traditional sense. Banks are institutions that provide a wide range of financial services, not just simply savings account as many people think. However, most people’s demand for banks is still mainly to deposit savings.
Yep, I agree with your assumption. I also think that Bitcoin won't be like a bank. My statements just supported your assumption.

Sure, banks is for saving. People are aware about it, they know that storing money into banks is a king of saving. They won't expect for profits.
When we store and fully control our own assets. This means the responsibility falls entirely on us. Whatever happen to our assets, we own it.
Indeed. we have the responsibility of our assets fully because we manage it ourselves. The decentralization system enables us to manage it in our own way.
Asset may increase or decrease in value. Asset as we know is something that one holds that has value. Although we wanted our asset to increase in value, an item that is devaluing over time can still be considered an asset. So not because the fiat deposited in banks is decreasing in value, it is not an asset anymore, it is still an asset called depreciating asset.
I'm not really sure what your point is. However, in my understanding, it can be called as an asset if it possibly raises its value. It is different with money that it always drops its value from time to time. So, I personally can't assume it as a kind of assets.