I have a hard time thinking of a situation where dual wielding would be the MOST appropriate response.
Really? I can think of lots:
1) Any one-on-one engagement where you've you've done more training in it than the alternatives.
2) Any situation where you're sufficiently trained in dual wielding, and carrying a shield is impractical due to size, weight, concealability, legality, finance and supply conditions, social custom, etc. For example, carrying a pair of knives is far more realistic in the vast majority of urban environments than carrying a sword and a shield. A ninja is not going to be carrying a body-sized shield on his stealth missions. A renaissance era gentleman can reasonably carry a parrying dagger into town. He's probably not going lug a tower shield around. He
could carry a buckler, but if he knows how to use it, an offhand dagger can not only be used to parry, but also to riposte.
3) Some specific weapons intended for specific applications. For example, both sai and hookswords have hooking prongs for which there are specific techniques intended to lock up your opponent's weapon while you attack him with the other.
4) Many short weapons exist that are pretty much always better to use in pairs. If you have brass knuckles, there are very few situations where it wouldn't be better to have two. Or consider sai and tonfa, for which a lot of fighting techniques exist that aren't really "weapon techniques" so much as empty handed fighting techniques but that are done with the weapons placed along your arm to act as forearm armor plus a punch range extender. You're pretty much always going to be better off with two of them than only one.
5) Some specific situations due to cultural and legal conditions. As mentioned above, for example, several Okinawan weapons are traditionally used in pairs. If you're an Okinawan peasant, you're legally prohibited from carrying a weapon. But you can legally carry farming tools that can be used as weapons. And some of them are simply better used in pairs. Again, sai being the best example. There's pretty much no situation where it's better to have only one.
6) Some situations where victory is hopeless, but you want to damage your opponent before you lose. For example, a peasant with kama attacked by a samurai is pretty much going to die. But if he has two kama he's far more likely to get in a stab or two before he does than if he only has one. Or, imagine facing off solo against the Mongolian hoard. Would you rather have one pistol, or two?
....and while we're at it, given the nature of this thread...
7) Any situation where you're not fighting in a real life army for your real life life, but rather you're engaging in non-lethal sport combat and
aesthetics is important to you. For example, imaging you're a young adult in a modern technological society that occasionally engages in simulated combat for fun. And imagine that you
want to dual wield for sheer awesomeness value. In such a
hypothetical case, it's probable that dual wielding and having fun doing it would be better than using a single weapon with or without a shield and
not have fun doing it.
tl'dr: shields are good for stopping arrows, but not all combat situations involve arrows. Dual wielding is generally inappropriate for army-scale engagements, but not all combat is army-scale engagement. Dual wielding generally takes a lot of training time to become proficient, so if you're taking your 10,000 conscripts to war next month, teaching them to dual wield is a bad idea. But if you're a rich dude with time, concerned about having a couple honor duels in your lifetime, you might be better off spending the time. And not everyone in every country over every period of time always has the option of choosing any weapon they want. There are cultural conditions and time periods where dual wielding has been more relevant than others.