Given South Africa's example, which has survived 25 years including orderly transitions of power (something few other regimes in Africa have managed) it is hard to see why it couldn't work. Switzerland also has a President who is a merged head of government and head of state who is not directly elected, and it is the paragon of stability, although the body that elects their President isn't a national parliament.
The foundation of most parliamentary systems were parliamentary systems in which the head of state was a sovereign who shared power with a legislative body in exchange for concessions from the people represented by the legislative body such as the right to impose taxes on the general public, and the right to raise armies via conscription. This is how constitutional monarchies arose, for example, in Britain, France, and Spain.
But, many modern parliamentary systems have an elected President serving the mostly symbolic residual roles of the sovereign head of state. Andorra has managed with co-heads of state who serve in other posts who are also heads of state ex officio (i.e. by virtue of holding another office). So, the South African solution differs mostly from the more common approach in Republics mostly by having a different way of conducting the election for head of state and removing a head of state, and by fusing the office of head of government and head of state.
The most important of those residual roles of a sovereign or President in parliamentary systems, from a practical operational perspective is the supervision of the formation of a government in circumstances in which no one political party commands a majority. But, the number of instances in modern history in which it would not have been possible to form a government when no one political party commanded a majority without the intervention of a head of state is few indeed (if there are any). And, even then, if the incumbent head of state serves until a successor is elected by parliament, as South Africa's constitution requires, that problem is largely overcome.
For example, in the U.S., Congress is vested with responsibility for resolving disputes over Presidential succession and disputed legislative elections, which are analogous to a situation in a parliamentary system where no one party can form a government and a head of state intervenes. And, in the few times where that has happened, legislative committees have resolved the disputed adequately for the United States government to continue.
In many countries the responsibility for resolving disputed elections is vested in a Constitutional Court instead of the legislature. South Africa, for example, has a Constitutional Court, which is an institution that was not available in early constitutional monarchies and Presidential republics where the head of state resolved constitutional crises. In India, their highest court has frequently resolved constitutional crises and this has also happened in the United States.
Another important role of the head of state is symbolic, and some of those duties are time consuming for a busy head of government. But, heads of government have some symbolic and political duties, and in countries with Presidential systems like the U.S. where the roles of head of state and head of government are fused, some of the symbolic duties of a head of state are delegated to others.
For example, in the U.S., someone serving as a first lady or de facto first lady (usually but not always the wife of the incumbent President), and the Vice President (famous for being deployed to funerals of foreign heads of state), and the Secretary of State appointed by the President with Congressional approval, all often share in the symbolic duties of the head of state. Even in constitutional monarchies, family members of the head of state are often deployed to carry out some of the symbolic rolls of a head of state.
Other examples of having offices that were traditionally either elected by the general public or hereditary appointed by parliament have worked well enough without difficulty. The House of Lords in the U.K. was traditionally entirely hereditary except for clergy, but has survived just fine with a large contingent of life lords appointed by parliament to the upper house. All of the seats in Canada's upper house, the Canadian Senate, are appointed by parliament again without doing any great harm to democratic self-government in Canada.
Many modern Presidential systems, among them, the United States, merge the role of head of government and head of state without incident. Other countries where the head of government is also the head of state include:
Angola, Argentina, Benin, Bolvia, Botwana, Brazil, Brunei, Burundi, Chad, Chile, Columbia, Comoros, Costa Rica, Cyprus, Dominican Republic, Ecuador, El Salvador, Eritrea, Gambia, Ghana, Guatamala, Honduras, Indonesia, Kenya, Kiribati, Liberia, Libya (with a legislative body elected President), Malawi, Maldives, Marshall Islands, Mexico, Micronesia, Nauru, Nicaragua, Nigeria, Oman, Palau, Panama, Paraguay, Phillipines, San Marino, Seychelles, South Sudan, Surinam, Switzerland (with a multimember federal council elected Presidency), Turkey, Turkmenistan, Uruguay, Venezuela, Zambia, and Zimbabwe.
While many of these countries aren't renowned for good government, the main distinction between them and countries with divided heads of state and heads of government is that those with the division usually had a monarch as some point (or still do) and are older. Fusion of the head of state and head of government is a relatively modern trend arising from the realization that the separation of the posts isn't really necessary.
(Prior to the adoption of the 1789 Constitution of the United States, its head of state was elected by a legislative body.)
Also, the role of giving someone a final say in resolving constitutional crises that is vested in the British Monarch and in monarchs more generally in constitutional monarchies, is handled quite adequately by the U.S. Supreme Court in the U.S. (something that wasn't possible in the U.K. due to the lack of judicial review and similar powers in the British system historically), by Constitutional Courts in many countries, and to some extent by the Council of State in France.
In communist countries, such as China, Cuba, North Korea, Vietnam, and Laos, the head of the communist party is the de facto head of state and head of government, elected by party members at a party congress, even though the head of state and head of government positions are formally filled by different people both selected by the communist party.
It is true that the head of state is harder to remove in almost every other political system, either because the head of state is hereditary or because the head of state is elected for a fixed term of office and can only be removed in extraordinary circumstances (e.g. impeachment in the U.S. or a somewhat similar process that is more trial-like in France). Arguably, that makes the system more stable.
But, when there is a stable Supreme Court or Constitutional Court to resolve constitutional crises in lieu of a monarch or mostly symbolic President, it is far less important that there be stability in the post of a head of state.
Given that every component of South Africa's system has been replicated with success in other countries, even though the exact combination that it has used has not been replicated, it is hard to see why it wouldn't work in the long run, or that it would be inherently unstable.
Put another way, there are many instances in history where a constitutional crisis has occurred and the governmental regime of a country has collapsed. Sometimes there are coups, sometimes there are revolutions or insurgencies, sometimes there are disputes concerning succession. But, I can't think of a single instance where the constitutional crisis led to the breakdown of a governmental regime, simply because the head of state screwed up the handling of the situation. There have definitely been instances when a head of state has smoothed over difficult moments in a country's political history, but it is hard to think of a single one where the performance or existence of a head of state who was not head of government was the make or break factor in that collapse.
Indeed, empirically, countries with directly elected strong Presidents are more likely to have coups than those with monarchies or those parliamentary systems, so having a directly elected President may reduce stability relative to one where a head of state is chosen by parliament, rather than reducing stability.
