How are we to make sense out of the information structure of Paul's argument at this point?
I think that the rules governing the use of "γάρ" are great guidelines, but I don't think it fits tightly into those rules. For when it comes to language: "Rules are meant to be broken"! So the force of "γάρ" may not always have the "supportive" nuance, especially considering variant readings often interchange "γάρ" and "δὲ". With that being said, I think its also helpful to consider some textual variants in this case. Generally, Byzantine readings replace "γάρ" with "δὲ" in verse 11, and add "γάρ" between εἰ and ἔτι in verse ten. In the Orthodox version then, it reads this:
10Ἄρτι γὰρ ἀνθρώπους πείθω ἢ τὸν θεόν ἢ ζητῶ ἀνθρώποις ἀρέσκειν εἰ γὰρ ἔτι ἀνθρώποις ἤρεσκον Χριστοῦ δοῦλος οὐκ ἂν ἤμην
11Γνωρίζω δὲ ὑμῖν ἀδελφοί τὸ εὐαγγέλιον τὸ εὐαγγελισθὲν ὑπ' ἐμοῦ ὅτι οὐκ ἔστιν κατὰ ἄνθρωπον· 12οὐδὲ γὰρ ἐγὼ παρὰ ἀνθρώπου παρέλαβον αὐτό οὔτε ἐδιδάχθην ἀλλὰ δι' ἀποκαλύψεως Ἰησοῦ Χριστοῦ
This is how I follow his arguments: with the initial "γάρ", It seems that paul is proving the point that he could never persuade God to change the gospel, the implication being that the Gospel ought to "persuade" men, not us to conform the Gospel to what we want. This is the theme of Galatians, that the Gospel is "ἐκ πιστεως" and not from "ἐργων του νομου." as the Judaizers were trying to push.
The Second "γάρ" in verse 10 adds a supporting statement to the implied answer, for the sake of the argument, that he is persuading men. Why? because if he is, his gospel would be false and he'd be disqualified by his own statement in the passage preceding verse 10 -- "ἀνάθεμα ἔστω"! This fits clearly in context, because in the first 2 chapters, Paul is defending his authority, and the Gospel he preaches.
In verse 11, we have a change in content (or a progression in discourse), Its the next logical step, the conclusion of everything said before, that his Gospel "οὐκ ἔστιν κατὰ ἄνθρωπον·" This is supported by the conclusion of his next "γάρ" statement with the referral of the ultimate source of his Gospel: "δι᾿ ἀποκαλύψεως Ἰησοῦ Χριστοῦ." This is an extravagant claim, and Paul knows this, so he leads the next verse (verse 13) with "γάρ"! He must bring forward evidence that Jesus Christ is, in fact, the ultimate source of his Gospel. He goes on about the change in his own life, and how he didn't hear the gospel from the apostles. He merely confirmed what he had received by revelation from Christ, when he finally met the apostles years later. It is only after this defense that he really gets to the point of his letter, and directly addresses the issue facing the Galatians.
One interesting note is the correlation of the thoughts in Verse 11 - 12, to that of verse 1. If "γάρ" is indeed the proper reading instead of "δὲ," Then maybe verse 11 - 12 is looking back on verse 1 with support. The issue with this though, is verse one talks about his apostleship, not his gospel. Either way though, the difference is very small.
so, with all that being said: