What the adoption curve looks like
Now, you're probably still skeptical of the potential for long-term adoption of cryptocurrencies as a medium of exchange, even with the TPS improvements, and with the radical innovation in price-stable coins. So below I'm going to paint a clearer picture of exactly how cryptocurrencies will come into mainstream use. Obviously there's a lot of speculation involved, but I really think there's a strong chance reality will follow a path similar to this.

A reliable USD-pegged or US treasury-pegged cryptocurrency will be created. The first step is for a reliable price-stable cryptocurrency to be created, backed by a group of legit investors. Some will argue that this already exists in the form of BitShares or USD Tether, but BitShares, in my opinion, doesn't have strong enough support in terms of its investors and dev team, among other issues, and USD Tether isn't really a cryptocurrency because it's centralized. It will probably take Sequoia or Google Ventures backing someone to make a price-stable currency really catch on, but I'm confident it will in the long run because the value proposition is so strong (read on).
A seamless way to convert from cryptocurrency to local currency will be set up. After a price-stable currency is created there needs to be some way to buy it using USD or some other local currency for people in other countries. This isn't hard to set up, and the team that builds the price-stable currency can just do it. Furthermore, cryptocurrency exchanges are generally happy to add currencies to their list of tradable instruments when they're backed by good investors.
Some country will try to inflate its way out of a crisis. The locals of that country will stop wanting to hold their country's tender. This is where things get interesting: I actually think the first people to adopt USD-pegged cryptocurrencies will be people in foreign countries . Although people in foreign countries generally hold their savings in the local currency, the situation changes when their country encounters a monetary crisis. During a crisis, particularly one in which the government becomes insolvent, central banks have an incentive to devalue their local currency in order to effectively forgive the debt their local government has incurred. This happened to an egregious extent in the Weimar Republic after World War I, but it happens today in many developing countries as well, and may even happen in some developed countries with high levels of government debt like Japan. In the past, the locals couldn't do much to escape their local currency's inflation because banks in their country couldn't hold USD or other inflation-resistant assets for them. With a price-stable cryptocurrency available, however, the game becomes much, much different. Anyone with an internet connection can go online and create a digital wallet that they can then use to hold cryptoassets. Yes, an exchange will need to be set up in their country for them to be able to buy the price-stable cryptocurrency, but cryptocurrency exchanges already exist in most countries, and they're much, much easier to set up than banks. Once acquiring inflation-resistant assets becomes easier, I expect the next country that tries to devalue its currency will see its population pursue a massive shift into cryptoassets. Seem far-fetched? Well, consider that people already flock to bitcoin whenever there's uncertainty in their local markets. The biggest consumers of bitcoin are the Chinese, and it's mainly because they're tired of all the games the government has been playing with the RMB for decades. The only thing stopping people from holding their savings in bitcoin during times of crisis is the price volatility. If you create a price-stable coin that pegs itself to something real, like USD in the short-term, you eliminate the largest barrier preventing people from piling into cryptocurrencies during a time of crisis. Yes, countries can try to impose capital controls that forbid people from buying cryptocurrencies-- but it's unclear to me that the legislative processes will move fast enough to stop the capital flight in time. If they don't legislate extremely quickly, local governments will face push-back from their populus, which won't want to lose its access to secure, inflation-resistant cryptoassets. In short: Once a reliable, stable cryptocurrency is created, I expect the next country that tries to inflate its way out of a crisis will see massive capital flight into this currency.
Payment processors will add functionality to process major cryptocurrencies. This is a bit more speculative, but once there's a sufficient number of people who understand the value of price-stable cryptos, before even any crisis occurs, I expect payment processors like First Data will start adding functionality to accept price-stable cryptos. The reason is that there's actually an extremely strong incentive for payment processors to do this because it allows them to stop relying on, and therefore stop paying fees to , its downstream partners: the payment networks like Visa/Mastercard and the issuing banks. Sure, people won't want to use this functionality at first because they won't want to forfeit their credit card rewards-- but once a crisis hits, the rewards won't be worth it compared to the inflation resistance afforded by price-stable cryptoassets.
People will realize there's no incentive to hold local currency at all anymore. Once you can hold your savings in a price-stable cryptocurrency and buy things locally with a price-stable cryptocurrency, the incentive to hold local currency disappears.