Interesting, isn't it? You did follow all that, didn't you? : )
Now, let's take a look at James 1:5 again:
αἰτείτω παρὰ τοῦ διδόντος θεοῦ πᾶσιν ἁπλῶς καὶ μὴ ὀνειδίζοντος καὶ δοθήσεται αὐτῷ.
ask God, who gives to all generously and ungrudgingly, and it will be given you. (NRSV)
So, according to Giovanni, the most salient (relevant) part of the noun phrase "God, who gives to all generously and ungrudgingly" is that he is "giving." The fact that he does so "generously and ungrudgingly" tells us how he gives, but James didn't consider that as important as the fact that our God is a "giving" God.
Isn't linguistics fun? : )