He answers his own question on his blog. I pretty much agree. It's interesting to imagine science fiction worlds where people do use volatile instruments like stock as their medium of exchange. Why would people in these worlds be willing to adopt volatile media while people in our world don't?@ClicheGuevara But why don't we use stock shares as our medium of exchange?— Noah Smith (@Noahpinion) June 13, 2015
Liquidity, the world's best insurance policy against uncertainty
First, we need to understand why our world has a preference for stable media of exchange. As Noah points out, people don't know exactly when they are going to need to spend money, or how much. Any individual faces the dizzying fact that any of an infinite number of events could hit them at any point in time. People have many ways to cope with this chaos, one of which is to build up an inventory of assets that can be deployed to help deal with surprises as they occur. Liquid assets—those that can be sold quickly, at low cost, and spent along multiple pathways, will do a better job of this than illiquid assets—those that take time to sell, incur transaction fees, and lack multiple pathways.
A buffer stock of liquidity offers people the same set of services as actual insurance, say a policy issued by GEICO. A home insurance policy immunizes against a range of disasters that might befall someone's residence. So does liquidity, since it can rapidly purchase materials and labour. If home insurance is cheaper than holding an inventory of liquidity (the cost of which, as Noah says, is an inferior expected return), then people will skimp on liquidity. Unfortunately there are only a limited range of future disasters against which people can purchase insurance. Good luck getting GEICO to insure you against a zombie outbreak, for instance. If a zombie scenario is something that you put a non-zero probability on, then staying a little more liquid than you otherwise would may alleviate some of your concerns. Come outbreak, the ability to rapidly dispatch liquid assets in all directions will come in handy.
Back to the question of volatility. If people are trying to build a moat against uncertainty, what use would insurance be if it offered $20,000 in protection on Tuesday, $17,000 on Wednesday, and $23,000 on Thursday? For the same reason that people require a fixed amount of insurance rather than a floating amount when constructing their moats, they will want to invest in stable liquid assets rather than volatile liquid assets. You can't solve for uncertainty with more uncertainty.
Liquidity is a virtuous circle
Liquidity is a virtuous circle; as an asset gets more liquid it becomes more attractive as an insurance policy, which brings in more buyers, which makes it more liquid, which increases its value as insurance, and so on. Once everyone holds the most-liquid asset(s), they have joined what in essence is an economy-wide mutual insurance scheme. At this point it makes little sense for a merchant to accept less-widely held assets as payment. Not only will it be a nuisance to set up the infrastructure, but the merchant runs into the coincidence of wants problem. Because the merchant's employees are already paid in the most-liquid asset, should the merchant accept volatile assets as payment he/she will have to bear the cost of converting them back into the standard unit. To avoid these inconveniences, only widely-held assets will be accepted by the merchant. A payments standard has developed.
Merchants will also try to please customers by setting their sticker prices in terms of the liquid asset. This reduces the calculational burden imposed on customers. At this point the asset has become a unit-of-account, and a pricing standard has developed. Adoption as unit-of-account provides all sorts of extra benefits to owners of the standard unit. As a courtesy, grocers and other retailers will typically keep their prices fixed for a few days, or weeks, which means that anyone who owns the standard unit knows ahead of time approximately how much food they'll be able to purchase. The tendency for prices to be sticky in terms of the unit of account only increases the standard unit's usefulness as a universal insurance policy.
The fact that shares aren't the unit-of-account means that consumers who own them miss out on all the uncertainty-alleviating benefits of sticky retail prices. They have no clue what their purchasing power will be one minute hence, let alone the next day.
Worlds with volatile money
If we lived in a world where we didn't need liquidity to shelter us from uncertainty, then we might be more receptive to using volatile media of exchange.
For instance, I sometimes wonder if the demand for dollar-denominated liquidity is less in Canada than in the U.S. since we Canadians have universal health care. With the nagging concern of how to pay for potentially life saving medical services solved for, we can economize on inventories of stable liquidity and seek out higher returns. Americans, who don't have such a product, probably need to hold larger dollar-denominated hoards in order to alleviate their queasiness about how they'll have to deal with future bodily harm.
Taking this idea to its limits, if an insurer were to introduce an "everything" insurance product (yes, this is science fiction), and it was cheaper than holding the standard liquid unit, then people would no longer need to self-insure against uncertainty by depending on the standard unit, typically central bank money and its banking derivatives. Instead, everyone would migrate their savings over to more volatile instruments like stocks, ETFs, or bitcoin, enjoying what Noah refers to as higher drift, or long-term returns. As long as GEICO is providing a low-cost universal salve against uncertainty, than people will be willing to shoulder all the inconveniences of fluctuating purchasing power insofar as it offers them a bit more drift.
With ownership of low volatility units (dollars, yen, etc) far less ubiquitous than before, and volatile asset (stocks, ETFs, bonds, etc) ownership much more prevalent, it might now make sense for merchants to incur the set-up costs of receiving volatile assets in payment. Furthermore, the fact that their employees will accept these volatile assets as salary, thus solving the coincidence of wants problem, will only accelerate the willingness of merchants to install the requisite infrastructure. After all, merchants can now pay their employees with the same ETF units that they receive from their customer, thus saving both commission expenses and the cost of incurring the difference between the bid and ask price. 
Information as a moat against uncertainty
In addition to liquidity and insurance, people can build their moats by acquiring more information. This is something I learnt from David Laidler. (If you want to learn about money via a fascinating detour through the history of monetary thought, you can't go wrong with these three books.) Knowledge allows individuals to better anticipate the future and plan accordingly, thus reducing the need for either liquidity or insurance contracts as hedges. If some cheap technology were to emerge that offered an infinite amount of free information (i.e. the internet?), then maybe people would cease amassing stable liquidity altogether. In this world, people's preferred solution to uncertainty would be to costlessly inform themselves and hold high return/high drift assets like stocks rather than stay uninformed and hold inferior, low return liquid assets.
Of course, we could also argue the opposite; that the internet has solved for one set of risks only to bring in a new set (identity theft, viruses etc), thus perpetuating the necessity of owning both GEICO insurance and liquid assets.
In sum, because our world lacks both infinite information and universal "everything" insurance policies, stable liquidity remain one of the cheapest ways to inoculate against uncertainty. The widespread prevalence of this monetary insurance across all strata of society has encouraged the development of a payments and pricing standards based on these assets. These standards further militate against the emergence volatile assets like bitcoin and shares as widespread media of exchange.
 I added this paragraph on June 19.