There is nothing inherently bad about mirror lenses. Like everything in photography a mirror lens trades off one set of advantages and deficits for a different set of advantages and deficits. For example, most mirror lenses have a fixed aperture. This turns the exposure triangle into an exposure line between ISO and Shutter Speed. On the positive side, this simplifies the photographer's decision making process by eliminating a variable. On the negative side, it reduces the photographer's options for rendering a scene into an image.
Short History
Mirror lenses for 35mm SLR's started appearing in the 1960's. They were constructed to very high standards by reputable manufacturers. Over time,
mirror lenses have continually moved down market. New mirror lenses are now among the least expensive lenses in the market irrespective of focal length. They are often less expensive than the least expensive "kit zooms" bundled with entry level cameras. Like "kit zooms," inexpensive mirror lenses meet a market demand for inexpensive lightweight compact lenses with useful focal lengths.
It wasn't always this way and good mirror lenses are good lenses. Compare the long focus throw of Nikon's 500mm f8 to that of a newly made Samyang/Rokkinon. Notice that the Nikon comes with a tripod collar while the new lens does not. Even if the optical quality of the Bower/Samyang/Rokkinon is good, it will be much harder to focus due to the short throw. The collar suggests using a tripod and using a tripod is likely to improve results. Bower/Samyang/Rokkinon lenses don't make the suggestion.
Another example of a high quality mirror lens is the Vivitar "Solid Cat" designed by Perkins-Elmer for military use and sold to photographers in the civilian market.
Challenges with long focal length lenses
Shooting long focal length lenses is can present challenges that may be at odds with a photographer's expectations. Some of those challenges are independent of the lens design.
Irrespective of lens type, any long focal length lens will have a narrow field of view. Locating and framing distant objects requires more hand eye coordination relative to shorter focal length lenses. Consistently acquiring and tracking moving objects at long focal lengths requires practice.
Long focal length lenses tend to produce images with shallow depth of field. Even at f8 a long lens will tend to isolate its subject. The hyperfocal distance of a 500mm f/8 lens on a 1.5x crop factor camera is approximately 1 mile (1600 meters). Most people will be shooting objects much closer than that.
Long focal length lenses are often used to capture distant subjects. There is more atmosphere between the subject and camera. Water vapor and air movement are more likely to effect images.
These technical challenges affect the experience of the type of photographer most likely to purchase a new Bower/Samyang/Rokkinon mirror lens. Most people find a 500mm focal length lens a more enticing purchase than the mid-grade tripod and head that will improve its performance.
Technical challenges of mirror lenses
Manual focus is more challenging than autofocus. It is a skill that requires practice. The shallow depth of field images produced by long focal length lenses make precise focus important. The short focal throw of the currently available new mirror lens designs makes achieving critical focus a more exacting task.
No image stabilization in the lens implies efforts to mitigate the effects of camera shake should be considered. Because system cameras from Nikon/Canon rely on in lens shake reduction exclusively this is relevant to many many photographers. 2
The design of mirror lenses is based on a mirror located on the front lens element. This front mirror blocks light from entering the lens and this means that there is more variation between fstop and tstop than is common with a conventional dioptric lens. An f8 lens will be probably be around T10. So there is less light.
The mirror on the front element creates ring bokeh. This can make distracting backgrounds more distracting particularly for photographers who struggle with selecting good backgrounds for their shots.
Technical advantages of a mirror lens
The mirrors can be very thin. This means that it is often possible to use less glass than a conventional dioptric lens design. Using less glass allows the a long focal length mirror lens to be lighter than a conventional dioptric lens.
By reflecting light from front to back, mirrors allow for a compact design. Compact relatively light lenses are easier to carry and to hand hold.
Mirror lenses can achieve close focus without a significant increase in complexity. Minimum focusing distances of less than two meters are common allowing "macro like" photographs.
Shooting a Mirror Lens
Using a tripod in conjunction with a live view on the LCD, is a good way to create sharp images in low light. Zooming into the live view lets the photographer evaluate the focus precisely. The technique is a little like ground glass focusing of a large format view camera.
With in camera image stabilization a mirror lens can be a "walking around" lens. Because the angle of view is approximates human foveal vision, composition is surprisingly easy: the detail that catches the photographer's eye is close to what the lens captures. The light caught my eye and all I needed to do was lift the camera and shoot from about 10m.
The long focal length allows filling the frame while maintaining a significant working distance between the photographer and subject. This is useful when proximity might change the subjects behavior. The distance provides a flatter angle than a close up with a shorter lens.
Relatively close focus provides macro-like photographs from working distances significantly greater than are typical when shooting with traditional macro lenses. This makes it easier for the photographer to avoid "standing in their own light" and casting shadows on the subject.
In general, a mirror lens offers and opportunity to show the world in a unique way. I had climbed half way up a narrow open fire tower to shoot a sunset. The compactness of the mirror lens meant it was on my camera and the camera was on a strap around my neck. I saw movement on the boat well over a kilometer away. It took a couple of seconds to focus and shoot.
The movement was pulling anchor. The shot is the person returning to their seat and ten seconds later the shot was gone.
Remarks
The boat shot defines the big reason to consider a mirror lens. I could not have climbed the tower with a conventional dioptric 500mm lens on my camera and my camera strapped around my neck. In the time it would take to unpack a conventional lens the shot would have been gone. The choice is not usually between a mirror lens and a conventional lens. It is between the opportunity to take shots that are not practical otherwise and no opportunity.
The bulk and size of a conventional dioptric 500mm lens means I might not have brought on the hike to the fire tower and the climb up it. Lightweight and compact, putting the mirror lens in my bag is not a particularly tough decision. It comes down mostly to weather.
For a mirrorless camera like the Sony A6000, older conventional dioptric lenses are alternatives to a new 300mm mirror lens. There are many very good 300mm manual focus lenses with an f4 aperture that can be mounted with an adapter for a price similar to that of a new mirror lens. They will most likely provide better optical quality at 300mm at the expense of size and weight.
The high Tstop of mirror lenses require more attention to the quality of light than might be required with other wider lenses. The constraints of mirror lenses reward the type of attention that translates to all other images. If nothing else, shooting a mirror lens will surface flaws in photographic technique, image planning, and the process of producing images. Some photographers may consider this a feature not a bug.
2: It is less of an issue for system camera with shake reduction in the camera body, such as Pentax, Olympus, and {sometimes} Sony. An overview of image stabilization current in 2017.