It definitely can stay both at the same time. It's just that with huge volatility present, the people who are going to use it for payments is just going to be a minority; pretty much like the current state of bitcoin right now whereas pretty much only the people in the bitcoin/cryptocurrency community are willing to pay using bitcoin. With time(probably a lot of time), bitcoin should attain stability as more regulations come in along with market liquidity. This is one of the things that are just a matter of 'when'; then bitcoin can be used and owned widely for both store-of-value and for daily payments(assuming bitcoin is very noob-friendly at that time).
I agree with what you say,and actually if I understand correctly, you are saying that it can technically stay the same, but not gonna work as well for both. Which is exactly my point. These 2 factors are counterproductive to each other. The more volatile, the less people will use it for daily payments. Because that means they are either loosing a lot of money money, or should hold it because they are earning money.
So I'm not sure if I got your idea correctly, but I think we agree on the fact that once it becomes stable, then it can become a standard payment method.
With all the user friendliness included for non-tech users.