I would agree vs I agree?
Solution 1:
You seem to have a handle on the grammatical differences.
In terms of word-choice motivation, it's used as a subtle way of distancing oneself from what follows ("I would agree that ...") through use of distancing language.
The "would" could also just be added out of a habit, or when one is trying to sound as if speaking at a higher register of language, as in the degree of formality of rhetoric.