Since the teleconverter only enlarges the final image, the image quality is highly dependent on the optical performance of the lens. If a lens renders an edge incorrectly, e.g. out of focus or with chromatic aberration, this can be amplified by the teleconverter.
The performance of the teleconverter also depends on the sensor of the camera. On a camera with a high resolution, image errors are much easier to detect than on a camera with a low resolution. So depending on the lens and camera, the performance of the teleconverter can vary greatly. In this article I refer exclusively to my experience with the TC1.4E III teleconverter from Nikon on my Nikon D850 and the 500mm f/4 FL ED.
The poorer image quality is not the only disadvantage of a teleconverter. You also loose light with a teleconverter. A certain amount of light no longer falls on the sensor and the amount of light for each pixel is correspondingly smaller.
With a teleconverter with a 1.4x magnification, you lose exactly one f-stop of light. The largest f-stop at 500mm f/4 is no longer f/4 but 5.6. With a 2x teleconverter you would even lose 2 f-stops. Instead of f/4, f/8 would then be the widest aperture.
To compensate for the loss of light, you either have to choose a slower the shutter speed or increase the ISO sensitivity. Especially with the increase of the ISO value, the image quality can again suffer significantly.
Another disadvantage of a teleconverter is the slower autofocus speed. Even though my lens is actually extremely fast, it had a lot more trouble with the teleconverter and was generally slower.
As you can see, the gain in focal length with the teleconverter is a big compromise. Not only do you lose image sharpness in general, you also lose image quality due to the lower widest aperture and therefore a higher ISO value. Finally, there is the slower autofocus, which in certain cases can also contribute to blurred photos. Despite these major disadvantages, it is not clear whether a teleconverter really makes sense. If you compare photos with and without a teleconverter, the photos are practically identical.