How are sentences like 日本がピンチだ (Japan is in a pinch) and 明日は雨だ (tomorrow it will rain) analyzed?
Translating them trivially, as
日本がピンチだ Japan is a pinch
明日は雨だ Tomorrow is rain
doesn't make sense. So something must be going on. Is it
日本 and 明日are topics, not subjects
This would only make sense if が is interpreted as a topic marker, which would be an expansion from its usual interpretation: a subject marker. "This wouldn't be a problem for the second example" you might say, but e.g. subordinate clause 明日が雨だったら is valid, and has the same problem.
ピンチ and 雨 are の-adjectives
ピンチの国 → 日本がピンチだ
雨の日 → 明日は雨だ
Loose/alternate interpretation of copula
You might argue that the Japanese copula doesn't equate things to the same degree as the English one, but merely associates them. This might seem the most straightforward explanation, but looking at the wikipedia article on copula, such interpretations are not mentioned.