It's not that it's too gory for Suetonius, because he describes the punishment elsewhere (as I'll get to). I think that an execution "in the ancient fashion" (antiqui moris supplicium in Latin at Claudius 34) is a figure of speech. He actually describes what it is in Nero 49.2 (thanks to Rolfe's Loeb edition of the Lives, I was able to find it). As Nero is on the lam at the end of his life, his freedman Phaon gets a letter in which he learns that Nero has been declared a public enemy, and that the Senate has decided to punish him in the "ancient fashion" (puniatur more maiorum is the Latin that appears here, in case you're curious). When Nero asks what this means, Phaon explains that they strip the criminal naked, put his head in a fork, and beat him to death with rods. Rolfe specifies that the fork (furca) is likely "two pieces of wood, fastened together in the form of a V" (p. 170 of the 1997 edition of the second Suetonius Loeb). Nero was apparently so terrified by this possible means of death that he immediately went for a dagger to commit suicide (but didn't succeed in killing himself without lots of complaining and histrionics, typical of Suetonius' Nero). Suetonius also mentions this "ancient punishment" at Domitian 11.2-3 (this time, more maiorum puniendi is the wording used). Domitian had actually condemned those charged with treason to this manner of death, but when he found out what it actually was, he was so horrified that he took back the punishment.

Looking back, it's kind of interesting that Suetonius seems to use this idea of an "ancient method of punishment" as if it's widely known, but the emperors themselves don't seem to fully understand what it is. There's a paper in there somewhere. Hope this helps!