For, say, a subtle background gradient to a web page, what are the advantages and disadvantages of saving it as a 16bit/channel png?
Are the colours represented by >8bits displayed in the intended detail in all browsers?
You have to make a value judgement - is it warranted for a background element? Probably not.
The advantage of a 8 bits per channel indexed PNG is its size.
The disadvantage is that if the image contains more than 256 colors, the graphics editing program will try to adapt the palette, meaning that some colors will slightly vary. Some programs are smart to pick a well-adapted palette. Some are retarded. Don't use the last ones.
For a subtle gradient, this disadvantage doesn't exist, since subtle gradients would hardly use more than 256 colors. To reduce the size of the image even more, you can restrict PNG 8 to use less than 256 colors: for example only 16.
Note about gradients in general:
Note about graphics editing programs:
Note about size of images:
Most users won't be disgusted by minor JPEG artifacts or PNGs with sub-optimal palette, especially on mobile devices. Speed, on the other hand, matters, especially on mobile devices. If a website is nice but every image takes ages to load because of its size, nobody will use the website. Consider this bias when choosing the appropriate format and settings. In other words, look more at the size of the image and less at the visual quality.
Some other hints:
Playing around with the images and especially PNG 8, you may discover the following:
¹ I don't remember the case when I've seen it. If somebody knows such a case, feel free to edit the answer.