Because destroyers arent meant to be used against battleships? They are meant to escort against everything and anything even if each individual ship had little stopping power they were meant to act in concert ot protect more important things.

Destroyers were in many ways the perfect small all vector surface combatant, at least how the USN developed and deployed them by the mid and late war. They could engage air threats with a variety of guns starting with the 5in main battery on all classes, along with smaller 40mm and 20mm mounts or 50cal mounts as well, utilizing radar fire control for the larger guns as the technology matured. Surface threats could also face the guns along with at least a handful of torpedoes in a trainable mount(though some ships gave up one or more mounts to add more anti aircraft guns by the end of the war. While subsurface threats could be met by sonar and a variety of depth charge and depth bomb devices.

So with the versatility of the destroyer established we need to understand how they were intended to be used, and that is in groups. The basic tactical unit was a division or 4-6 vessels, these could be supported both administratively, and logistically as a unit and would typically be shifted around from task force to task force as a group as well. Multiple Divisions would then constitute a Squadron, which at that point 8-12 destroyers starts to be a real force to reckon with, enough to engage in independent actions somewhere, or provide sufficient screen for a carrier or two backed up by a handful of larger cruisers or maybe battleships. Nor were more ships in the screen ever unwelcome, especially after experiences in the early war saw much more adhoc protection. AT Midway for example the USN was able to sortie between the two carrier task forces 3 under strength Destroyer Divisions for a total 15 DD's. While 2 years later at the Philippine Sea each USN carrier task force, centered around a pair of fleet carriers, was protected by at least a dozen destroyers each!

Nor was it just the carriers that needed protecting, the Fast Carrier force needed dozens of tin cans for escort work but so too did the large invasion fleet units in the Amphibious Train coming up behind, along with independent task forces operating elsewhere in the Pacific from the 7th Fleet in Australia, to supply and reinforcement convoys across the vast Pacific. Along with some to protect key bases like Pearl itself.

And while destroyer escorts, which were built by the bushel as well, could take much of this load, destroyers were usually present in at least small numbers with most convoys or rear area units to provide some additional weight to the screen. Many of the DE's (about 100) were also built and served only briefly with the USN before being transferred to the Royal Navy on top of the old flush deck destroyers transferred before US entry into the war, with other forces such as the Free French Navy also getting handfuls of ships of various classes.

The last notable caveat I would point out is that the chart and video is semi meaningless, as it doesnt point out that many of the ships being constructed either never served against Japan or split their time in both Europe and the Pacific. For instance at 3:09-3:10 the Fletcher class ship, the bad luck William D. Porter and Hall show up. Both of which spent their early time in service on the East Coast, notable escorting FDR aboard the Iowa to the Tehran Conference, before serving in escort and anti sub duties either off North Africa from Gibraltar, or in the Atlantic out of Bermuda. Then in late 1943 both were ordered to the Pacific, where Porter sank and Hall survived the war.

The need to fight a truly global multi ocean war was the driving factor for many USN procurement decisions, when you had to be able to provide escort to convoys from the Philippines, the enormous wastes of the Pacific, India and SE Asia, Europe, Africa, and to the USSR, along with protecting the long East, Gulf, and West coasts a lot of escorts were called for.

To add on to u/DBHT14's excellent answer, destroyers were also important for several other duties as well, some of which he has already alluded to.

1. Radar pickets. The threat of kamikaze aircraft lead Commander John Thach and several other U.S. Navy officers to create the "big blue blanket" defense scheme for carrier task forces. At all times, a combat air patrol of Hellcats and/or Corsairs would orbit above the fleet. These fighters would be guided to incoming threats by fighter director officers on the carriers. These officers used the carrier's own radar to spot these threats. But sometimes a carrier's radar couldn't see the threat in time for the CAP to react or more fighters to be launched.

To give the fleet more early warning, the "big blue blanket" system called for destroyers and destroyer escorts to be placed up to 50 miles away from the rest of the task force. These ships would act as radar pickets, giving the task force early warning of incoming attacks. However, these pickets were horribly-exposed and often suffered merciless attacks,

USS Laffey (DD-724) was hit by four bombs and six kamikazes off Okinawa and nearly sank. Today, it's gone to a very well-deserved rest as a museum ship.

2. Anti-submarine warfare - destroyers and destroyers were in constant demand for sub-hunting in the Pacific and the Atlantic. This included convoy escort and patrol missions, as well as participation in more active sub-hunting missions. Destroyers and destroyer escorts were also part of "hunter-killer" task forces, where a small group of surface escorts would work with an escort carrier to find and sink enemy subs.

In other cases, the escorts worked without a carrier. The destroyer escort USS England (DE-635) and two other destroyer escorts managed to sink 6 Japanese submarines in just 12 days of May 1944. Acting with help from Allied codebreakers, England sunk a Japanese supply sub and then "rolled up" a picket line of Japanese subs.

3. Shore bombardment - Destroyers took part in every major amphibious operation of the war. During the D-Day landings, destroyers like the USS Emmons (DD-457) braved German guns and mines to get within 1,000 yards of Omaha Beach to fire their 5" guns at German positions killing G.Is. on the beach.

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On a side note.

Because destroyers arent meant to be used against battleships?

Broadly speaking, this is true! Destroyers were certainly never intended to replace battleships in U.S. Navy service.

However, I would add a major caveat, since U.S. Navy doctrine and practice did call for destroyers to engage enemy battleships when possible. Specifically destroyers were to make night torpedo attacks prior to the battle and daylight torpedo attacks during the height of the fleet engagement.

Trent Hone writes about this in this aspect of 1930s-1940s U.S. Navy "major tactics" Naval War College Review article "U.S. Navy Surface Battle Doctrine and Victory in the Pacific."

Cruisers and destroyers were drilled in “night search and attack” procedures, designed to locate and damage an enemy battle fleet the night before a battle.

...

Coordinated attacks using planes, destroyers, and the guns of battleships were a recurring feature of the Navy’s Fleet Problems and tactical exercises.

There are multiple cases of destroyers being used in line with this doctrine. The most notable example being Jesse Oldendorf's destroyers at Surigao Strait in October 1944, which make night torpedo attacks in the Japanese ships transiting the Strait prior to Oldendorf's battleships opening fire. One of the tin cans even manages to torpedo and sink a Japanese battlewagon!

Hone again:

By noon on 24 October, Kinkaid had realized that the Japanese were seeking major action and ordered Oldendorf to prepare for a night battle in Surigao Strait and to take Berkey’s TG 77.3 in support. Oldendorf began to formulate a plan. He called Berkey and his battle-line commander, Rear Admiral George L. Weyler, to his flagship so he could familiarize them with the details. He also discussed the use of destroyer attacks before the main action; he planned to send them down both sides of the strait.

The radar pickets at Leyte Gulf and onwards also soaked up a number of kamikaze attacks that, had they not been there, would have been directed against support vessels, carriers and other larger surface combatants. In the horrible calculus of warfare, replacing a few destroyers is cheaper than the other ships*. The destroyers weren't helpless, as by this time the excellent VT radar fuze and later model fire control directors made their AA fire very deadly. To define this further, the kamikaze pilots were being provided a highly mobile and defended target that was cheaper to replace than say, the USS Intrepid or an ammunition ship and so forth. Some of the less desperate kamikaze pilots ignored the the picket ships and went for the fleet, of course.

• I personally knew one veteran who had two ships sunk under him at Leyte. TWO.

There are some great breakdowns of the USN AA data here. http://www.ibiblio.org/hyperwar/USN/rep/Kamikaze/AAA-Summary-1045/index.html