There’s certainly a lot of ego in his statement, and I agree that he either entirely or mostly means himself in the second branch. But to me it’s the first branch that is the difficult part to understand, while the second is straightforward.
Leaf’s note on this, surprisingly to me, elaborates on the μυστηριάζεσθαι idea.
ἐφ’ ὑμείων, as T 255 επ’ αὐτόφιν εἵατο σιγῆι. The idea seems to be, ‘Do not let the Trojans hear your words, lest they may endeavour to counteract your petitions by prayers of their own’; this he immediately revokes by the καί in 196, virtually = nay. There was a widely-spread primitive idea that every local or national god could be approached only by a particular form of words, which was therefore carefully concealed from an enemy. Thus the title by which the god of Rome was to be addressed was concealed, as a state secret of the highest importance. So the real pronunciation of Y-h-v-h was kept secret by the Jews, Jehovah being only a conventional form for reading with the vowels of Adonai. It is said that the direction in the Prayer Book to read the Lord’s Prayer ‘with a loud voice’ goes back to a period when this too was superstitiously regarded as a magical formula to be repeated silently, lest the enemies of Christianity might find it out. 195-9 were athetized by Zenod., Aristophanes, and Ar. on the ground that ‘they are not consistent with the character of Aias, and that he raises objects to himself (ανθυποφέρει ἑαυτῶι) absurdly’; a judgement which does not comment itself. It is certainly not inconsistent that a hero, after recommending a conventional precaution, should correct himself, and say that he has no need of such devices. (See, however, Wilamowitz H. U. p. 244.)
Wilamowitz’s referenced argument:


Now, I don’t think there’s much internal support for Leaf’s and Eustathius’s idea of secret prayers here. But it very much reminds me of Pratum Spirituale story 196, in which some shepherd boys have overheard the secret words of mass, and engage in a mock celebration. (Compare also Alcibiades and the profanation of the mysteries.) The ancient world certainly had some interesting ideas about prayer secrecy that we don’t share today.
Notice the part I’ve bolded below, and compare to Leaf’s claim about the Lord’s Prayer at one time being a magical formula to be repeated silently.
Διηγήσατο ἡμῖν ὁ Γεώργιος ὁ τῆς Ἄφρων χώρας ὕπαρχος γεγονὼς, ὁ φιλόχριστος καὶ φιλομόναχος καὶ φιλόπτωχος, ὁ πᾶσι χαίρων τοῖς ἀγαθοῖς οἷς ἀγαπᾷ ὁ Θεός. Ὅτι ἐν τῇ χώρᾳ μου (ἦν δὲ τῶν μερῶν Ἀπαμείας τῆς δευτέρας τῶν Σύρων ἐπαρχίας)· ὅτι χωρίον ἐστὶν ἀπὸ μιλίων τῆς πόλεως τεσσαράκοντα, τὸ Γοναγὸν καλούμενον. Ἐν τοῖς ὄρεσιν οὖν τοῦ αὐτοῦ κτήματος ὡς ἀπὸ μιλίου ἑνὸς παιδία ἔβοσκον θρέμματα. Καὶ οἷα συμβαίνει, καὶ ὀφείλει γίνεσθαι παιδίοις, ἠθέλησαν παῖξαι κατὰ τὴν τῶν παίδων συνήθειαν. Καὶ ὡς ἔπαιζον, λέγουσι πρὸς ἀλλήλους· Δεῦτε, ποιήσωμεν σύναξιν, καὶ προσενέγκωμεν προσφοράν· καὶ τούτου πᾶσιν ἀρέσαντος, προεβάλοντο ἐξ αὐτῶν ὡς ἐν τάξει πρεσβυτέρου, καὶ ἄλλους δύο, ὡς ἐν τάξει διακόνων. Καὶ ἔρχονται εἰς μίαν πέτραν ὁμαλήν· καὶ γὰρ ἔπαιζον. Καὶ ἐπὶ τῇ πέτρᾳ, ὡς ἐν τάξει θυσιαστηρίου διεθήκασι τοὺς ἄρτους, καὶ ἐν καυκίῳ ὀστρακίνῳ οἶνον. Καὶ παρίστανται, ὁ μὲν ὡς πρεσβύτερος, οἱ δὲ ὡς διάκονοι, ἔνθεν καὶ ἔνθεν. Καὶ ὁ μὲν τὴν προσκομιδὴν ἔλεγεν, οἱ δὲ τοῖς φακιολίοις ἐῤῥίπιζον. Εὑρέθη δὲ ὁ ῥηθεὶς πρεσβύτερος τὴν ἀναφορὰν ἐπιστάμενος· ἐπειδὴ καὶ ἡ συνήθεια ἐν τῇ Ἐκκλησίᾳ παρέδραμεν, ὥστε τοὺς παῖδας ἔμπροσθεν τοῦ ἁγίου ἱερατείου ἵστασθαι ἐν ταῖς ἁγίαις συνάξεσιν, καὶ πρώτους μετὰ τοὺς κληρικοὺς τῶν ἁγίων μεταλαμβάνειν μυστηρίων. Ἐπειδὴ δὲ ἔν τισι τόποις ἐκφωνεῖν μεγάλως εἰώθασιν οἱ πρεσβύτεροι, εὑρέθησαν τὰ παιδία, τὴν εὐχὴν τῆς ἁγίας ἀναφορᾶς ἐκμανθάνοντα, ἐκ τοῦ συνεχῶς αὐτὴν ἐκφωνεῖσθαι. Ὡς οὖν πάντα πεποιήκασιν κατὰ τὴν ἐκκλησιαστικὴν συνήθειαν, πρινὴ τοὺς ἄρτους μελίσωσιν, πῦρ ἐκ τοῦ οὐρανοῦ κατελήλυθεν, καὶ τὰ προσκομισθέντα πάντα κατέφαγεν, καὶ τὴν πέτραν κατέκαυσεν ἅπασαν· ὡς μηδὲ τὸ παράπαν μήτε τῆς πέτρας, μήτε τῶν ἐν αὐτῇ προσενεχθέντων καταλειφθῆναι μνημόσυνον. Καὶ τοῦτο ἀθρόως γενόμενον θεασάμενοι οἱ παῖδες, εἰς τὴν γῆν ἀπὸ φόβου κατέπεσαν, καὶ ἡμιθανεῖς μεμενήκασιν, ῥῆξαι φωνὴν ἢ ἀναστῆναι ἐκ τῆς γῆς μὴ δυνάμενοι. Μὴ εἰσελθόντων δὲ αὐτῶν εἰς τὸ κτῆμα τῇ ὥρᾳ ᾗ εἶχον ἔθος εἰσέρχεσθαι, ἀλλ’ ἐκπλήκτων κειμένων εἰς τὸ ἔδαφος, ἐξῆλθον οἱ τούτων γονεῖς ἐκ τοῦ κτήματος ἰδεῖν τίνος χάριν κατὰ τὸ ἔθος οὐ παρεγένοντο. Καὶ ἀναζητοῦντες, εὗρον αὐτοὺς κειμένους, μήτε τινὰ τῶν παρόντων γινώσκοντας, μήτε λαλῆσαι τοῖς λαλοῦσιν ἰσχύοντας. Οὕτως ἡμιθανεῖς αὐτοὺς οἱ γονεῖς θεασάμενοι, ἕκαστος τὸ ἴδιον παιδίον βαστάσας εἰς τὸ κτῆμα ἀπήνεγκεν· καὶ ἐν ἐκστάσει τοιαύτῃ τὰ τέκνα ὁρῶντες ἐξίσταντο· τὴν τῆς ἐκστάσεως αἰτίαν μαθεῖν μὴ δυνάμενοι. Καὶ πολλάκις αὐτοὺς δι’ ὅλης τῆς ἡμέρας ἐρωτῶντες, ἀποκρίσεως παρ’ αὐτῶν οὐκ ἐτύγχανον, καὶ τῶν συμβεβηκότων αὐτοῖς οὐδὲν ὅλως μαθεῖν ἠδυνήθησαν, ἕως οὗ ἡ ἡμέρα ἐκείνη πᾶσα καὶ ἡ νὺξ παρῆλθεν. Τότε γὰρ οἱ παῖδες εἰς ἑαυτοὺς κατ’ ὀλίγον ἐλθόντες, ἄπαντα αὐτοῖς διηγήσαντο, ὡς ἐπράχθη τε καὶ γεγένηται. Καὶ λαβόντες αὐτοὺς οἱ γονεῖς, καὶ τοὺς τοῦ αὐτοῦ κτήματος κτήτορας, ἐξῆλθον, καὶ τὸν τρόπον ὑπέδειξαν, ἐν ᾧ τὸ παράδοξον τοῦτο θαῦμα ἐγένετο, ἴχνη τινὰ τοῦ καταβάντος πυρὸς ὑποφαίνοντες. Οἵτινες τῶν γεγονότων ἀκούσαντες, καὶ ἐξ αὐτοῦ πεισθέντες τοῦ πράγματος, εἰς τὴν πόλιν εἰσέδραμον, καὶ πάντα τῷ ἐπισκόπῳ τῆς πόλεως καθ’ ἑξῆς ἀνήγαγον. Ὁ δὲ θαυμάσας τῶν λεγομένων τὸ μέγεθος, ἐν τῷ τόπῳ σὺν παντὶ τῷ κλήρῳ ἐξῆλθεν, καὶ τοὺς παῖδας ἰδὼν, καὶ ἐπ’ αὐτῶν ἀκηκοὼς τὰ γενόμενα, καὶ τὰ σημεῖα τοῦ οὐρανίου πυρὸς θεασάμενος, τοὺς μὲν παῖδας εἰς μοναστήριον ἔβαλεν, τὸν δὲ τόπον εὐαγὲς μοναστήριον πεποίηκεν· ἐπάνω τοῦ τόπου τοῦ πυρὸς τὴν ἐκκλησίαν ᾠκοδόμησε, καὶ τὸ ἅγιον στήσας θυσιαστήριον. Ἔλεγεν δὲ ἡμῖν ὁ αὐτὸς κῦρος Γεώργιος, ὅτι ἕνα ἐξ αὐτῶν τῶν παίδων ἦν θεασάμενος ἐν αὐτῷ τῷ μοναστηρίῳ, ἔνθα τὸ τοιοῦτον θαῦμα γέγονεν, καὶ τοῦτο μὲν τὸ θεῖον καὶ ἀγγελικὸν θαῦμα διηγήσατο Γεώργιος ὁ φιλόχριστος.