Why do some viruses cease being a problem even though no vaccine or cure is found?
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.