Nice is the poster boy of semantic shift.

Yes, it is. But, in the case of nice, its older meanings have, more or less, been replaced by its current meaning. There are some other outcomes. At least two off the top of my head: (a) words like set and file that have multiple meanings but are basically from the same older word (etymologically speaking) and (b) words like mole which has multiple meanings and has almost as many etymologies.

1. Hand me the file.
2. Mary was out standing on the mole.

1. and 2. are ambiguous, except that sentences like these are extremely unlikely to occur in complete isolation. 1. depends on whether we are in a metal shop or a doctor's office; 2., more humorously, depends on whether it's uttered by a dermatologist, zoologist, or harbor master. Context, and in speech, repetition and emendation, usually remove any ambiguity from the text.

Next, the argument that polysemy leads to the eventual failure of communication and the destruction of the language is easily proved wrong by looking, however cursorily, at a page in a dictionary (in any language): there are very few entries for words with a single meaning, and those are usually words that a rare or archaic.

Ceci n'est pas un seing.