If you're willing to impose a certain level of realism on the show, then such a fleet would indeed be absurd by WWII-era naval warfare standards.
In WWII, the biggest source of firepower for a navy was large ships, primarily battleships and aircraft carriers. Such ships required a huge amount of resources to operate - Kaga, Zuikaku, and Kongou each had well over a thousand personnel on-board. They could be used effectively in a variety of situations, from naval combat, to blockades, to bombarding enemy troops anywhere near water.
But, such large ships had a fatal weakness, which had been known since the 19th century: torpedoes. A battleship could easily sink weak enemy boats from a long distance away. For example, Kongou's 356 mm guns had a maximum range of 35 km (the effective range was somewhat lower, but still quite far). However, if the enemy forces had a large number of small boats, each armed with torpedoes, it would be difficult for the battleship to take down all of these with its relatively slow weaponry. Certainly, it could hit some of them, but with enough numbers it wouldn't be able to take them all down effectively. Aircraft carriers fare a bit better in this respect, but they too had to worry about any sort of armored torpedo-carrying vessel or submarine, and enemy fire could prevent aircraft from launching or landing, which would basically cripple their offensive capabilities.
With enough ships, the enemy forces could easily advance to within the effective range of torpedo fire (which depended on the model of torpedo, but was around 4-8 km for one common torpedo used by the U.S. Navy). It would not take long to deal heavy damage to the large ship, possibly even sinking it. Small enemy vessels (such as cruisers) could also quickly approach to get in range of their own guns, at which point the main advantage of battleships and aircraft carriers (long range) is lost. If this seems like a poor strategy, keep in mind that torpedo boats could be very small (at some points even converted merchant/fishing vessels were used for this purpose), and operated by only a few people, yet a small number of them would pose a serious threat to much larger battleships and aircraft carriers. Thus, the big ships need some way to deal with smaller ships approaching them too quickly for the large ship to deal with.
This is the reason destroyers were invented. Destroyers were relatively heavily armored (given their small size), highly mobile small ships which were primarily used for short-range tactical combat and to escort larger ships. While torpedo-boats and other swarms of small ships could out-maneuver large ships with enough numbers, they weren't a threat if you could intercept them before they got in firing range and take them out. A destroyer like Fubuki would generally only be used in combat against other naval forces. Their medium-caliber guns could still easily outrange (maximum range of over 18 km) torpedoes, and would be a threat to approaching cruisers as well. A destroyer only required around 200 personnel, and more than just destroying enemy torpedo boats, they were generally pretty versatile: they could be used for holding tactical positions, reconnaissance, torpedoing enemy ships, attacking enemy submarines (essentially just underwater torpedo boats), and anti-air attacks, depending on the needs of the mission and the available equipment. Even in the worst case scenerio that the destroyer sinks, this still buys time for the (much more important) large ships to escape, fight back, or wait for reinforcements.
Generally, any larger ship during WWII would be escorted by at least one destroyer. Especially important ships might get more than one destroyer escort. So, for a fleet of 2 aircraft carriers, one battleship, and 2 light cruisers (Kitakami and Ooi are described as torpedo cruisers, but this is just a type of light cruiser armed primarily with torpedoes), we'd expect at least 5 or so destroyers, and possibly even more. For a real-life example, the Japanese fleet at the Battle of the Eastern Solomons had the following numbers, which were reasonably typical for a battle of this size:
- 3 carriers
- 1 seaplane tender (essentially a light carrier)
- 2 battleships
- 16 cruisers
- 25 destroyers (more than the above 22 ships combined in raw number)
- Several smaller/non-combat boats
A fleet of 2 carriers, 1 battleship, 2 cruisers, and 1 destroyer is rather absurd in comparison; it's putting far too many resources in the large ships, and not enough into protecting them. A swarm of small ships would pose big problems for a fleet like this. While such a fleet would have excellent offensive capability given the number of ships, they wouldn't have especially good defensive capabilities, and would be an attractive target for enemy attacks (basically a glass cannon).
In addition, the flagship was the most important ship in any operation. The commanding officer of the operation was on the flagship, and crippling or sinking the flagship would not only damage enemy morale, but also their communications and strategy. Making a ship that was designed to be relatively expendable, like a destroyer, would be a bad idea. The enemy forces would likely all target the destroyer, and the fleet would be in serious trouble if it sank, usually forced to retreat.
Typically, the flagship was one of the largest ships in any given operation, being difficult to sink, well-defended, and powerful enough to boost the morale of other ships in the operation. A small ship being the flagship would probably have problems with respect as well as being an easy target. While there were a couple cases where a destroyer was used as the flagship for an operation, these were generally only in cases that the whole fleet consisted of only destroyers, or when the original flagship was too heavily damaged to continue (such as in the Battle of Guadalcanal).
With all that said, none of this really applies very well to the Kancolle universe, at least in the game. Destroyers had a predominantly tactical, defensive role, but the game's battle system doesn't do much to emulate this. In game, most of the advantages of destroyers (evasion and speed) aren't particularly unique to them; one can replace destroyers with larger ships without much loss. This will incur larger ammunition and fuel costs, but it isn't necessarily a bad strategy or one that the enemy forces will have any way to exploit.
Indeed, once one progresses far enough in the game, even more unbalanced fleets become commonplace. Later missions require more and more firepower, which is increasingly hard to meet with weaker ships like destroyers and cruisers. The game is also generous enough with resources that the fuel and ammunition consumption become relatively unimportant. Since the number of ships you can bring to a battle is limited (as are your enemy's, preventing any kind of swarming strategy), it's often better for high-level missions to only deploy battleships and carriers. If you're well prepared, it's relatively rare that such large ships will take any serious damage, and with some strategy there's essentially no risk of them sinking. If anything, the single destroyer would be the weak link defensively, and unlike in the real world, where losing a single destroyer in an otherwise good battle would be seen as a big victory, in the game you can't afford to throw away experienced destroyers regularly.
Making the destroyer your flagship would offer it some protection, since in-game flagships can't be sunk, but if it reaches a heavily damaged level (which is quite plausible) you'll be forced to retreat. There are other benefits for the flagship, such as increased leveling and a chance to have attacks hit other ships instead. There are strategic benefits to both large and small ships being flagship, but in any case it's not crazy to make a weak ship like a destroyer the fleet's flagship.
I suspect there's no good in-universe reason to consider this strategy crazy or impossible, at least based on what we know now about battles in the Kancolle universe. In the game, it wouldn't really be at all, given the right circumstances. The anime hasn't presented us with any explanation, and it has an original story (since the game has almost no story), so we can't really find any answer there. But in the context of actual WWII-era naval battles, it would indeed be a crazy, illogical decision to make a fleet like this, and there are no historical examples of such a fleet.