Why is cobalt-57 radioactive? Why isn't it stable? Cobalt-57 should be the most stable and common isotope of cobalt, right?  It has three more neutrons than protons....
Nickel-58 only has two more neutrons than protons, and it is the most common/stable isotope of nickel....
Why, precisely, is it so susceptible to 'electron-capture'?
 A: On the LibreTexts website the page Isotope stability contains information relevant to the question. 
A nucleus is classified as stable when there is no known half-life
On the page a diagram labeled 'Type of decay' is displayed.
For each atomic number (the amount of protons) there is an optimum for the ratio of neutrons to protons.
When the number of neutrons is smaller than that optimum there is a decay mode where the number of protons decreases.
When the number of neutrons is larger than that optimum there is a decay mode where the number of protons increases.
According to the diagram: for the entire range of atomic numbers the 'strip' of isotopes that has no known half life (black squares indicating stable nuclides), is a single line. That is: in the diagram there is nowhere an area with four black squares in a 2x2 square
It seems exceedingly unlikely to me that the narrowness of that strip is a coincidence. What's going on?
Here is what I think:
There is a general rule that in any physical system with freedom to rearrange itself towards various states the system will end up in the state of lowest energy, simply because that is the most probable outcome.
As long as a lower state of energy is available there is a non-zero probability that some process brings the system to that lower state.
If the current state is the lowest possbible state of energy then there isn't anyting left to decay to.
An isotope having a very long half-life means the decay mode that brings the system to the lower energy state is an unlikely event to happen. For instance, it may be that the decay event has to overcome a large 'energy hump'. That reduces the probability of the decay event happening, but it cannot be reduced to zero. 
Concluding:
I can't begin to guess why Cobalt 57 specifically has a half-life, but the above general considerations seem plausible to me.
