I won't into too much analysis of the debate--other than my suspicion is that Obama was leading by three touchdowns, played a prevent defense, yielded a field goal, and took some time off the clock--but there were a few (policy) opportunities that the President missed.
I'm not going to worry about the question of whether he should have gone off more aggressively on the 47% issue, or on Mitt's taxes or offshore accounts, or accused Romney of lying. I'm just going to focus on policy.
I'll be a bit more wordy than Obama gets to be--it's hard to say anything substantial in a two-minute debate. (And really--the moderator should have "off" buttons for the microphones). But you get the idea.
Tax policy
A lot focused on a good tax policy, and the favorite Republican position that lowering taxes causes economic growth. Romney scored a point when he noted that Obama essentially conceded as much with the payroll tax extension--don't want to raise taxes in a recession; and we're still in one. Obama didn't respond effectively.
Which is unfortunate, because the effective response is obvious and illustrates a key difference between the candidates: Not all tax cuts are created equal.
Putting more money into the hands of the poor and middle class--whether through stimulus programs, tax cuts or whatever else--does improve the economy. People like us (I'm quite middle class) spend money when we get it. Or retire debt. Even if we save it, it increases our ability and willingness to spend later as we enjoy more financial security.
The payroll tax cut was targeted at the bottom four quintiles--which is why Republicans hate it.
Romney, on the other hand, wants to focus tax cuts on the top percentiles--the "job creators" in Republican doublespeak. While a good argument can be made that reducing marginal rates in exchange for eliminating deductions is generally a good idea, cutting taxes on the rich does far less to stimulate the economy; this is especially true when you are talking about income taxes--levied on what is left over after a business pays its bills. And if you are financially secure, additional money is less likely to materially change your consumption habits.
Obama should have made the point--forcefully--that tax cuts for the rich often end up being a windfall for the rich, and don't stimulate the economy. Or stimulate overseas economies, as more and more production moves offshore.
Debt reduction
Romney said that one means of debt reduction would be to "pass things on to the states". Presumably, he means that certain programs would be ended at the federal level, and the revenues previously used to fund them would then be used for other purposes--paying down the debt, or financing tax cuts. The problem, though, is then states would have to raise revenue to do these things; and the state governments are far harder up for cash than Uncle Sam--most of 'em can't borrow money except under specific circumstances, and none of 'em can borrow at rates near 0%.
Shifting programs to the states is not a magic formula. Even if one presumes state governments can do things more efficiently (a dubious proposition), it's just passing the buck.
Role of government
Romney gave the usual conservative schtick about free markets good, government bad, federal government especially bad. Obama responded with a few mostly-irrelevant anecdotes about the government being useful sometime and somewhere. If there is one damaging conservative talking point that remains an albatross around the country's neck, it is that this belief in the inherent inferiority and inefficiency of government, at least in economic matters.
The response is simple: Private actors are looking out for themselves, not for you. Government agencies are run in the public interest, or at least they have a duty to be (regulatory capture is a big problem, but the solution is not to throw the baby out with the bathwater). Competitive markets work great in many cases, but many essential services are difficult for the private sector to efficiently provide--without either wasteful duplication or monopolistic rent seeking.
And of course, the main reason that conservatives like to privatize is that they dislike progressive taxation. The more things are privatized and taxes reduced, the more net wealth transfers to them, instead of the other way around.
Health care
Here is where Obama did best; as Romney--being the architect of Romneycare, the template for ACA--has to thread a needle here, disavowing his own creation without looking like a total opportunist (or the heartless bastard many believe him to be). And he flailed quite a bit--making a blatantly false statement that was walked back by his campaign as soon as the debate was over. Romney also made a federalism argument--that this should be an area of state concern (and that RomneyCare/ACA is fine for Massachusetts or any other specific state, should they choose, but not for the US); and doubled down on an asinine proposal that John McCain advocated in 2008--creating a national insurance market.
Obama should have shot that down. He did shoot it down in 2008 against McCain, pointing out that insurance companies would simply move to the state(s) with the least regulation under such a regime. And he could even note that federal pre-emption of state regulation of insurance would be a type of nationalization all its own. (It has long been my opinion that there are very few true federalists in DC; most politicians support nationalist or states-rights positions on issues depending on who is in charge where).
That said, I'm not sure that this one matters as much. The people mainly motivated by federalism are probably already likely to vote for Romney; for many swing voters, this is an abstract issue that doesn't put food on their table. (But it certainly gets the Villlage up in arms...)