No vaccine or cure has been found for SARS-CoV or MERS-CoV yet the world is not too concerned about them. How can it be? Did all people who had those viruses die or do viruses just randomly decide to stop being infectious at some point?
Infections spread in a population when the number of new infections caused by an infected person is greater than or equal to 1.
If each infected person spreads the virus to less than 1 person, eventually no one will be infected, without a need for any sort of cure. Of course, the longer a deadly infectious disease spreads in a population, the more people will die during this process. Ideally you want to get the number of transmissions as close to zero as possible as quickly as possible.
SARS and MERS are each a bit different, but share some similar characteristics. Compared to COVID-19 caused by SARS-CoV-2, both are more severe in a larger fraction of the people infected (note: all three viruses are closely related betacoronaviruses). This made it easier for public health officials to identify and isolate infected individuals.
The MERS virus is really not gone at all: it lives on in animal reservoirs, like camels (so it is primarily only a concern in places where those reservoirs live). Cases continue to occur sporadically. However, MERS is usually not that transmissible between people, having a natural transmission rate that is already less than 1: that means that most people who get it get it directly from an animal, and don't continue to spread it to others besides occasional infection of individuals with close-contact, like a family member. There have been exceptions where isolated incidents involved substantial human-to-human transmission. Some of these incidents were associated with "super-spreaders": particular individuals who got infected and spread the virus to way more people than the average. See https://wwwnc.cdc.gov/eid/article/26/2/19-0697_article for more.
SARS caused by the original SARS-CoV virus indeed seems to be gone, with no cases reported since 2004. The elimination of SARS from the human population occurred via controlling the human-to-human spread through isolation and contact tracing. See https://apps.who.int/iris/handle/10665/70863 for a report on the epidemic and how cases and spread in different locations were handled.
Viruses don't have any agency: they can't decide to do anything. However, humans can, and our best response to outbreaks of novel diseases is to trace the spread and try to limit transmission as much as possible.
Bryan Krause's answer addresses the reasons pertinent to SARS and MERS. If you meant those two as examples but are interested in the title question more generally, I can note an additional mechanism. This is herd immunity, which fits the bill in that it can occur when "no vaccine or cure is found" and when it is neither the case that "all people who had those viruses die" nor "viruses just randomly decide to stop being infectious". After a significant fraction of the population has been infected and recovered, herd immunity can bring a virus with $R_0 > 1$ to an effective rate $R < 1$ if those that are recovered are immune to reinfection and this immunity lasts for long enough.
And considering the title more broadly,
We learn to live with them, manage the infections, and consider a significant death toll or other related injuries as "normal".
For example, a measles vaccine was only developed in the 1960s. Before that, it was just known as one of the diseases children would usually get when they were young, because the virus was endemic in the population. On average, 2.6 million people a year died of measles globally, and that was just expected. (For comparison, that's now down to 122,000 by the way, and most of them are in extremely poor countries who either can't afford the vaccine or can't reach clinics.)
For an alternative case, the rate of congenital disabilities or birth defects in children was simply accepted as what it was - something which just happened without any cause. It was not until 1941 that the connection was made to rubella (German measles) causing up to two-thirds of all birth defects. Before that point, we didn't know that this wasn't "normal".
And currently of course we still have no solid vaccine for flu. The vaccines we do have are relatively effective and certainly do reduce the impact, but they are not effective to the same extent as, say, the MMR vaccines. As a result, we still have thousands of people dying of flu every year, and currently that is "normal". Maybe one day we'll crack it, but until then we'll just expect people to continue catching it, doctors and hospitals to continue treating it, and some people to die from it.
One more effect: a pathogen that is both deadly and highly contagious either triggers strong response (when we talk about humans, human pets, cattle, etc) or simply wipes out the population it infects (and disappears with it).
This creates a strong selection pressure over the pathogen. In order to successfully spread, it either becomes less deadly, or less contagious, or both. The generations of the pathogens change quickly and mutation rate is usually high, so the pathogen (given time and groups of hosts to infect) will react to the selection pressure.
The groups, once survived the less-deadly version of the pathogen are at advantage - they are most probably immune also to the original. The pathogen is also at advantage - it gets less organizational reaction (quarantines, vaccines, etc) and more hosts to infect (because survived hosts reproduce and the new hosts are not immune).
p.s. a second-order of the same effect is the pathogen becoming less dangerous only to minors of the host. We all know a lot of examples.