I'm arguing that two assets having the same expected return should be priced the same no matter how much their actual return deviates from their expected return.
Both expected returns are the same so maybe more people will choose the risk free option in that case. But their actual return will be different in the scenarios you've provided and variation alone can cause the pricing to be different (explained next).
It is captured by the "beta" in the formula
The beta only standardized the risk but doesn't necessarily factors in the volatility. It only provides insight to systematic risk but does not factor in non-diversiable risk that is specific to 2$ coin toss that 1$ does not have.
rational investors would still be better off buying volatile assets and run away from risk-free assets promising a lower return.
Maybe in the long run, holding risky assets MIGHT provide a higher return if the market is good. But on the other hand the more volatile asset will lose more if market is bad. eg.
Comparison between stocks and bonds. Stocks should have a higher expected return and more volatile as bonds are usually treated as risk free assets. When times are good and everything goes up, stocks out performs bonds. But as soon as market is bad, bond outperforms stocks.
I know but that's irrelevant to my discussion as risk in finance is always calculated in the same way (volatility).
It's pretty relevant since other models incorporate volatility and risks whereas CAPM doesn't. For example, the popular Black-Scholes model has sigma in addition to risk rate in order to factor in volatility.
My argument is that volatility is not how risk should be calculated.
Fine, tell us how it should be calculated then. Honestly, I don't even know what to say anymore since this is literally basic finance stuff.