Actually, he may have purposely NOT written anything, just to show the "scripture alone" crowd who's boss.
The Catholic faith, as we all know, is composed of more elements than just scripture. Scripture, says scripture, is hard to understand, cannot be taken any way we like, and doesn't even contain everything that Jesus said and did. It also supports the Church, the sacraments, the Church's authority, etc...
The Church recognizes tradition to be another very key thing for salvation. It contains many things, I'm sure, that are not in the Bible. Yet that doesn't mean Christ never said or did them, or commanded us to do them.
A very good bishop once pointed out in so many words, that God's way is not to pound people over the head with a big hammer of truth, so that the truth is so obvious they must either completely accept it, or reject it and go all the way down to the lowest pits of hell, because they knew absolutely that it was true, and rejected it totally and willfully. He pointed out the example of the dead sea scrolls, wherein was found one little scrap that did not match anything at all of the Old Testament. Now those who do not want to know, could surely dismiss this as just the incompetence of the researchers to find what part of the old testament it "really belonged to". But the fact of the matter is, it DID match completely with some part of the NEW Testament, thereby debunking (for those not utterly blind) that long-held error, "oh, the gospels weren't written until hundreds of years AFTER Christ, and they were just pius inventions and religious hysteria of a few sentimental men" (or words to that effect).
The Bishop pointed out, Christ COULD have given them a whole page sealed up in that cave. He could have given them a whole book of the new testament, and it could've been the book they thought was older than any other, to boot, to REALLY shoot their error down. But he didn't. He gave them one, tiny scrap. Now those who are sincerely in error, but with a good will to hold the truth, will not be QUITE so guilty of not believing the truth. For those who know the truth, that scrap was kind of like a winking confirmation, I suppose.
You're right... it IS very interesting that Christ did not WRITE anything, in spite of having no qualms about showing that He knew everything, even as a young boy in the temple. But it's really interesting that the fact that He DIDN'T write anything seems to be a very big nod to the necessity and importance of tradition... AND... to His promise that He would therefore preserve that tradition precisely because it IS that important. I personally can't think of a better way He could have done this, in fact, other than to simply not write anything. Think about it. If you believe in scripture alone and call yourself a Christian... you do not have one, single thing that Christ, as a man, physically wrote. You have what other people wrote that He SAID... but you don't have what He wrote with His own hand, and scripture says, "this isn't ALL He said and did." SO... there could be then, things that Christ commanded, said and taught, which are NOT in scripture.
Of course the Catholics just nod and say, "Yeah... that's why we have tradition." But the scripture alone crowd has got this serious, glaring flaw, and this is it. If the Bible doesn't tell them everything Christ said and did, how do they know they have all the info they need to do everything He wanted them to do, or everything they have to do to save their souls. Only tradition can tell us the rest of the picture, and that means God will have had to preserve it, and flawlessly, for all men of all times, in some form that ISN'T scripture.
Of course, if Christ had written by His own hand, explicitly, every point of our faith or something, then everyone who didn't belong to it, or didn't want to, would be that much more guilty, and have to suffer that much worse in hell if they go there, for having rejected it. But it seems like this interesting fact is really a big nod to the Church. Remember, "he who hears you, hears Me"? I would guess that is why we don't have any writings of Christ. He founded this Church, and meant for men to listen to it. If they're so stubborn and blind that they won't hear those whom He sent in His name, I'm sure they would have found some excuse not to believe the writing of Christ Himself.
Anyhow... This is just a really lovely example of what's wrong with the whole "scripture alone" thing. Thanks a million for pointing this out, too, because I'd never thought of this before. A very, very interesting point, and something to perhaps bring up in conversation with someone I know who is in the "scripture alone" camp.