There may well be more emphasis on a note rather than a chord. As mentioned at the end of your question.
Modal interchange and parallel keys are both used regularly in pieces, which says one particular note rather than a chord is given more weight.
Pieces which use only diatonic notes will more readily be recognisable as being in a key - or two. There's always the relative major/minor waiting in the wings.
Given your example of C, G and Em, there's an ambiguity with them. All three could belong to key C, key G, or key Em. A bigger clue could be when G is followed directly by C - the perfect (authentic) cadence is probably the most used - V>I, possibly putting it into key C. But that's by no means an acid test. Until we know what other chords come after, that's only a vague guess.
Then there's the fact that modes of one key will all contain the same notes (and thus chords) as each other - giving seven possible 'answers'. Here, the constant return to either a particular chord or note usually signifies the 'key'. So the answer is either or both. Obviously emphasis caused by where particular notes/chords are used in a piece will also have a bearing on the clues. A simple example may be if every 4th bar, the same note/chord is used woud point to that being 'home', therefore the root note/key.