Are prefixes, as bound morphemes, always separable from their root words?

Prefixes are morphemes that are attached to root terms to create a new or more complex concept. However, the root term can have many variations, and a given word that contains a prefix may not make sense if the prefix is removed.

Your example of promote is one of these. The following is the explanation of the etymology of promote from the Compact OED

late Middle English: from Latin promot- 'moved forward', from the verb promovere, from pro- 'forward, onward' + movere 'to move'

Many prefixes were already attached in the language from which the word derived. This is an example. Movere (to move) is the root and pro- (forward) is the prefix. Promovere was an acceptable term in Latin before it migrated to Middle English and modern English.

Some of the derivative words in English can stand alone when the prefix is removed, such as promotion (pro + motion = forward + movement).

Even if the word was created in modern English by adding a prefix to a root, the variations on the word may not allow the root part to stand alone when the prefix is removed. For example, telephony is a term that was created in modern English. If the tele (distance) prefix is removed, the root part phony cannot stand alone. However in telephonic, when you remove the prefix, phonic can stand alone.