No, there's no flaw in his logic. Comparing the two religions to determine which is "worse" is a silly and pointless exercise. Any religion can be perverted to an evil purpose quite easily. The scriptures and theology of every religion contain much than can be used to justify any sort of behavior.
To the extent that there is worse behavior coming from Muslims in the present day world, that is most likely due to differences in culture. Religion follows culture much more than culture follows religion. If we posit that 200 years from now, the Muslim world has undergone a liberal enlightenment, I suspect that somehow Islam will seem like a much more tolerant religion than it does today. Look at the behavior of Christians 500 years ago. What changed? The Bible is the same.
Sure, any religion can be perverted, but so can any other point. This isn't an argument of "LOL EVERYTHING CAN BE GOOD OR EVERYTHING CAN BE EVIL DEPENDING ON HOW YOU LOOK AT IT LOLOL," it's an argument of: do the majority of their followers agree with their actions, or do they view them as a crazy subsect independent of the whole? As a baseline, does the rest of the world view this subsects actions as fucking crazy? The answers to both of these questions must be identical for an agreement to occur. These agreements do not occur because, as I said, there's a far greater percentage of Muslims that do shit like this than Christians.
It makes absolutely no difference whether or not they have undergone "liberal enlightenment." As you yourself state, sure they may be "more tolerant" in 200 years, but the fact of the matter is that this is not 200 years in the future. Sure, five hundred years ago Christians did some fucked up things they wouldn't do today, but that was five hundred years ago -- Muslims do them in the present, which is all that matters.
You don't see the majority of the world saying "Hanukah? Mother fucker those people are outrageously crazy," or "Man, they refuse to kill even a single bug? If that fucker didn't kill some bug I saw, I'd kill him," but you do see the majority saying "Wow, they fucking stoned some woman to death for being raped, then gave the guy who raped her a medal. That's fucking crazy."