While the dictionary entries you cited are technically correct, I think they are missing the sense of the word, which I think is what really explains it. I work in programming, and deal with a similar distinction at work, where "=" "==" and "eq" all have nuances of the same basic meaning, 'equals'.
To say two things are equal implies actual, qualitative similarity. Equal means two things ARE the same, at least with regards to the quality you're talking about. Being coequal, rather, is a more general statement about the relative standing of the things being considered.
All people have equal rights. But you and your boss are not coequals, there is a difference in standing.
From your example, if you said that all three branches of government were equals, hits the ear funny, equals with regards to what? You can say they have equal power. You can say they have equal standing. But they are coequal.
To back this up with a reference, here you go:
https://wikidiff.com/equal/coequal
This cites a similar difference as I point out above. Equal points to actual equality, while coequal says the things are "equal to each other in size, rank or position."
Best Answer
We need a nominal (a word that can take on noun functions) to act as the complement of the copula is. Both the infinitive ("to verb") and the gerund ("verbing") serve this purpose, and provide the same grammatical meaning in this particular context.
To my native ear, the gerund sounds smoother, but the infinitive sounds more formal.