Yet wouldn't the Pope put himself at risk at having a counter-Pope or an ambitious papal representative stationed in Jerusalem seize it for himself. He would have the advantage of claiming to be Christ's true vicar on Earth ruling the Holy City in His name instead of the Pope in Rome. As for the drain of manpower, I wouldn't think so since there is land elsewhere to take in Egypt and Syria.
Anti-Popes didn't spring up willy-nilly like rebellions in a game of Medieval: Total War, and the Crusades didn't happen in the 20th Century. Simply being based in Jerusalem wouldn't give someone the legitimacy to claim to be the true Pope. You're thinking in modern day ideals. Anyone sent by the Pope to rule Jerusalem and then suddenly claiming the right to rule all Christendom would be laughed out of court by every King in Europe. Nowhere in the Bible does it state that Jerusalem is the centre of Christianity. As a matter of fact, the Crusader state of Jerusalem had a Patriarch of Jerusalem, but at no point did this Patriarch claim to be superior to the Pope, it's just a ludicrous assertion. To do so, apart from anything else, totally defies the Church principle of submission to the Pope - to separate oneself from the Pope's power, even for a second before declaring yourself Pope, is a precedent which unspokenly says that the Pope's power can be subverted if you believe it just, and that's just totally out of the question. The only time Anti-Popes arose in Christian history (ignoring the idiotic ones in the last couple of decades RL) were in times of turmoil, when there was a very controversial election and ambitious Bishops took their opportunity to call for a re-election which conveniently got them elected. Never was there an assertion that Rome wasn't Holy enough for the Popes. For that matter, aside from being the Pope's home since the 1st century, there was nothing really Holy about Rome anyway in the way that Santiago de Compostela or such was Holy, yet it never caused problems.
At very worst, what you would see is the Patriarchs of Jerusalem becoming so prestigious that they almost automatically win every Papal election - becoming an heir to the Pope, like the title King of Germany was the heir's title to the Holy Roman Empire. But it's frankly anachronistic to think that any Patriarch based in Jerusalem would start declaring himself the REAL head of the Church. It just defies all of Catholicism's doctrine of Papal supremacy, and instantly destroys the position that it is trying to subvert.
On the other issue, of a muscle-drain instead going to Syria or Egypt...perhaps, but what makes you think that a Papal Jerusalem would become so successful as to enable a Christian conquest of Egypt? That would take a total fluke victory in my opinion.