I'm a little confused here. You say that bitcoin and fiat have no value for direct use, with which I agree. You then say that fiat does not work as a store of value. But then you imply that bitcoin can be a store of value that will suck in all of the extra stored value of the housing market. Why wouldn't bitcoin and fiat be the same in this regard? Yes, bitcoin has limited supply, but why does that mean that it should be considered a safe store of value (it would have to be considered even safer than real estate to suck in its stored value)? Or are you thinking of a future in which bitcoin basically becomes the reserve currency of the world, or at least the dominant currency of a particular country or region?
The difference between fiat and bitcoin is precisely that fiat is unlimited, and bitcoin is limited in supply. Since fiat is unlimited in supply, it can not be used as a store of value. Each time the supply is doubled, the stored value is halved (well, after the new fiat is dispersed). This is the main difference that bitcoin makes. It is in effect now, no need to wait until all others use bitcoins, if they ever will.
In one sentence, you say that fiat can't be used as a store of value, and then in the next, you talk about the stored value of fiat being halved. So can fiat be used as a store of value or not?
I think fiat can be, and is, used as a store of value. The stored value does fall over time with inflation, but it is still used as a store of value. When you save money, you're storing something of value that can be exchanged for goods and services.
If bitcoin stabilizes in price and becomes an ingrained global currency, then its limited supply might make it a better store of value than fiat. But for now, fiat is a better store of value because of its stability.
Fiat is a store of value, but not a good store, as the value evaporates with time. Other money, with a near stable value, is better. Bitcoin is more risky as it can go down further, but since it can also appreciate, it can be described as stable with volatility, or stable with a risk. But I expect the value to go up as far more likely than to go down, so as a store of value it is far better than stable money.