If we have a series system with $n$ components (so if any of the components fails, the system fails), the failure rate of the system becomes simply the sum of the failure rates of the components. It doesn't depend on the mean time to repair/ repair rates of the components at all. This is apparent in example 9.32 of the book "Introduction to probability models" by Sheldon Ross (10th ed). It has a mechanical derivation of this, but I don't have an intuitive feel.
I can see why if the repair times are very small, the overall failure rate is simply the sum. But if the first components time to recovery increases, won't some of the transitions to down for other components start falling into the periods when the first component is already down? So, shouldn't this reduce the total transitions to the down state of the system, resulting in a lower failure rate than the simple sum of failure rates across components?
It is a single point of failure system. The system fails when any of the components fail.
You cannot repair a component until it fails.
Therefore the failure rate of the system cannot depend on the repair rate of the components.