During a period like this one, when the president is backed by a majority at the parliament, he runs the show. It means generally that he makes the major decisions and decides about the great political orientations but let the government in charge of the actual implementation of the policies (excepted foreign policy and military issues that traditionnally he follows more closely).
There are two main reasons for this : the first one is to that the president is suposed to stay a little above petty political controversies and more of an ultimate arbitrator (way much less so than in traditionnal parliamentary democracies, though), and the second, more important one is to avoid exposing himself too much. He generally lets the prime minister takes the heat, and if there’s too much popular dissatisfaction, or if a major change of orientation is decided, he picks a new prime minister, like he did recently by sacking Raffarin. Generally speaking it worked quite well in the past to prevent presidents from becoming too unpopular.
However, in situation of crisis, the president is expected to become more publically proactive (normally, he is very active, but not too publically). But in this case, as you noted, he has been conspicuously absent. I’m not sure why, as I didn’t hear or read any rumor about him being for instance (as you mentionned) hampered by health issues. And making some public speech from time to time shouldn’t be that difficult anyway, barring a really serious condition (his predecessor, with a cancer in terminal phase at the end of his mandate, was in all likehood unable to fully fulfill his charge, but still showed few signs of it when he appeared in public.
My guess would be that he doesn’t want to be too much involved in the catfight between Villepin and Sarkozy (especially since the current situation is presicely directly related to Sarkozy’s hobby-horse issue : public order and safety), maybe hoping that Sarkozy (whom he deeply dislikes) will fail, or will become unpopular enough, or will be the target of too much criticism, so that it will become politically possible to get rid of him. The former president Mitterrand used such a tactic with someone who was similarily his arch-rival within his own party, Rocard, putting him in charge, offering him no support altogether, and letting him “fail on his own merits” to get rid of him.
But it’s just a guess. Maybe the pesident just wants to avoid taking blame in such a volatile situation and his PR advisors thought he would be best if he doesn’t say too much.
Another possibility would be that he’s keeping himself “in reserve” in case the situation would turn really bad, and he would need to takes the reins directly and put as much moral and political weight in the scale as he can, rather than being already discredited following a prematurate and failed direct involvment. Which is very possible, given that it would be a behavior similar to his role-model De Gaulle’s, who thought that the president must keep his political capital intact so that he could be the ultimate recourse in case of a major crisis (like the Algeria war, or the 1968 riots).
I would note, though it’s an extremely remote possibility, that the french constitution (article 16) allows the president to take on essentially dictatorial powers when the situation is such that the regular functionning of the political institutions is disrupted (it was implemented once during an attempted military coup at the end of the Algeria war). To do so, once again, the president must still have an intact authority and a strong moral capital. Despite the likehood of the situation detoriorating to such a point being extremely low, he might have this possibility in mind too.