I've only read modern retellings on the siege of Carthage, and they all have one thing in common; They mention how Rome demands the Carthaginians hand over their arms, which they do... Rome then demand they move the city inlands and they refuse, starting the war.

I don't get it. How did the city hold out for 3 years without weapons. Did they get new ones or something? Modern historians have a tendency of skipping this detail for some reason.

Ineffectively?

That is a good question. My guess is that the historical sources are not clear on that. I think it comes from ambiguous writing, which was not uncommon in ancient texts. People weren’t as exact as we are now. So the phrase “The Carthaginians gave up (all) their weapons.” Could mean that a Carthaginian army gave up most of their weapons. I put the word “all” in parentheses because it is possible that the all was added during paraphrasing or translating. Even today, people add an “all” when paraphrasing others.

One way to look at it, is that after WWI Germany was forced to disarm, but was still able to have a small military force. Of course, Germany secretly built more weapons as well as found ways around the size limit for the army. Wikipedia cites Goldworthy to say that records indicated that the Carthaginians gave up “200,000 sets of armour and 2,000 catapults” as well burnt their warship. I am not sure if the records mention things like spears or bows and arrows, and Wikipedia leaves it out, or if they were left out of the treaty. Anyway, that is a lot of armour. Estimates of Carthage’s population range from 90,000 to 800,000. Also, an estimate for the Carthaginian force was about 20,000. So it is possible that the Carthaginians just sent a lot of armour, that wasn’t actually counted, and the Romans couldn’t really verify if it was all the armour, as they were in Utica. It seems also possible, that small arms were overlooked, but maybe sets of armour includes helmet spears in shields. I think it is a safe assumption that the Carthaginians kept some weapons, if it was because the demand to disarm was not meant to be that strict or if they tricked the Romans, is another question.

Carthage could probably also produce their own weapons, and probably ramped up production even before the actual fighting started. I found this site, which claims that "In this period, the 500,000 Carthaginians inside the wall transformed the town into a huge arsenal. They produced about 300 swords, 500 spears, 140 shields, and over 1,000 projectiles for catapults daily."

I don´t know much about it, but maybe there is some actual scholarship on it that someone here knows of.