I'll preface this by saying most of this comes from Goldsworthy.

Rome's refusal to negotiate peace terms was an anomaly for ancient Mediterranean warfare. You're correct that Hannibal never really had the resources to siege Rome itself, but he likely felt it wouldn't be necessary. Hannibal had won stunning victories against the Romans (Trebia, Trasimene, and Cannae) that should have brought the Republic to it's knees based on the losses alone. But Hannibal didn't stop there. He actively worked to turn Rome's allies against her, with many allies who resented the Roman yoke joining Hannibal. He campaigned around the countryside, wrecking Rome's agriculture and economy.

By any measure of the time, surrender should have been forthcoming. Any other state would have considered itself soundly beaten, and the prospect of an enemy band roaming their lands would have brought most other nations to the negotiating table. But not Rome. Rome simply refused to give up. I don't think Hannibal anticipated this course of action and thought his many victories would have forced Rome to concede to Carthaginian demands.

WRT your question about Hannibal's overarching strategy, Lazenby makes a cogent argument that is, to me, the most persuasive. If Philip V of Macedon could understand that the secret of Rome's success was to absorb other peoples, then it is not unreasonable to expect the same of Hannibal. Couple this with the fact that he surely would have known that any war with Rome was going to be long and drawn out- the First Punic War had lasted twenty three years, and when faced with catastrophic losses, the Romans had simply thrown more men and resources into the war. If Hannibal stayed on the defensive, the Romans would keep sending armies at Spain and Africa until they won; equally, the history of Roman diplomacy tells us that climbing down over Saguntum would not have brought peace- not for long, at any rate. Long term, the only real option was war, and the nature of Roman power ensured that the only viable strategic option was an invasion of Italy, where, instead of a war of Carthage's resources versus Rome's, Hannibal would be making use of Rome's resources for Carthage and denying them to Rome.

If the secret to defeating Rome was detaching her from her allies, then that dictated fighting the war a very certain way and undoubtedly limited Hannibal's freedom of action. In his History of Warfare, Montgomery laid into Hannibal for essentially not laying siege enough, but this misunderstands what Hannibal was doing- with bringing the Italians and Latins into the fold as the goal, he could hardly set about assaulting Brundisium or Beneventum (Lazenby notes that Hannibal's only assaults appear to have been for strategic purposes, like acquiring a port). Once people did come over to him, he then also had to protect them from the inevitable Roman retribution, further limiting him, especially when concluding peace on terms such as he did with Campanians.

Concerning the question of marching on Rome after Cannae, consider the situation. Rome is over 250 miles from Cannae- Hannibal's army normally marched at a pace of about 9 miles a day, maybe 10-15 on a fast march. It would have taken the main force a bit more than Marharbal's five days to get there, and given everything else we know about the Romans, it is highly unlikely they would have just thrown open the gates in despair and surrendered. The city still had substantial defences: two legiones urbanae, 1,500 men at Ostia, the marines Marcellus had stationed at Teanum Sidicinum, and on top of that, all the able-bodied citizens, many of whom would have already seen combat, and the armed slaves. Hannibal had taken noteworthy losses during Cannae and a lengthy siege would have left him vulnerable to being surrounded by relieving forces (gaining the city through treachery, as he did in other cases, is pretty inconceivable here). His forte was pitched battle, not sieges (see how long Saguntum held out). Regardless, if the peoples of (particularly southern) Italy were to be persuaded to defect, how would they feel about his disappearing north to the capital? They had stuck to "wait and see" policies after the Trebia and Trasimene, and may have done so again without the Carthaginian army remaining.

The ultimate failure of this strategy does not, I think, make it a bad one- indeed, Carthage's ultimate defeat obscures the fact that it was working! Lazenby calculates that by 212, over 40% of the allies were no longer available to Rome. Even Latin loyalty was, according to Livy, considered uncertain by the Romans. As late as 209, twelve of the thirty Latin colonies refused to supply their contingents of troops. Other allies were (again, we only have Livy's word for it) equally unhappy and thought that enough of them were disgruntled enough, they might persuade the Romans to negotiate a peace with Hannibal. Think of the manpower losses Rome suffered: by the end of 216, something in the area of 120,000 men had been killed or taken prisoner, and there were problems with the levvies in later years.

I will finish by quoting one of Lazenby's final paragraphs in its entirety:

"I still think Hannibal fought [the Second Punic War] as well as it could have been fought... when after seven years a strategy has resulted in the defection of the two largest cities in the enemy's country, after the capital itself, along with over 40% of the enemy's allies, and when the enemy army has been comprehensively defeated three times, on the last occasion perhaps suffering worse casualties in a single day's fighting than any other army in European history, it can hardly be regarded as a complete failure."

Lazenby, J., "Was Maharbal Right?" in "The Second Punic War: A Reappraisal" (Institute of Classical Studies, 1996)