The Proxima class may earnestly be called the definition of the word Overkill. Sporting more weapons than any ship prior and a lot of ships since wherever the ships went, at least two others had to follow as support. As for why Starfleet constructed this absolute beast of a vessel, that's certainly a story.
The Proxima class stems from, of all ships, the Federation class technology-wise. Whilst the Federation had been an unstable failure of a vessel the ships were often used for engineering experiments, primarily aimed at stabilising the ship's dual-warp core setup. In 2262 a breakthrough was finally achieved and, whilst the Atlas had already superseded the Federation as the fleet battleship, the two Federation class ships were at least of sufficient quality to enter regular service guarding the Sol System. It was this breakthrough that allowed thought to move to a successor to the Atlas class, one using technology derived from Starfleet's latest unreasonably successful technology testbed, the USS Miranda.
The design slowly moved forward with the project for the Proxima class, which ballooned in size to 375 meters in length. Sporting two pairs of the now proven Chesapeake M1 nacelles the Proxima was expected to support an astounding 14 FH9 Phaser arrays. This was supported by 6 torpedo launchers and two shield emitters, powered by the same SSWR-XVIII warp core as the Oberth class. Whilst that may sound suboptimal, this problem was solved by simply using four warp cores, split across two tubular hulls. Whilst this wasn't as unstable as the launch configuration of the Federation class it still took a good few alterations to get the USS Proxima running properly, allowing for extreme power at the expense of astoundingly high antimatter consumption.
The USS Proxima launched in 2269 to noted fanfair, and whilst the ships proved popular with the press their reputation among the fleet proved rather negative. The quadruple warp core caused concern and, more concerningly, frankly ridiculous fuel consumption. Even the worst of the Eaves-Kurtzman ships, the Walker Class, burned only half the antimatter of the Proxima, and the space taken up by the warp cores meant there was less than enough space for fuel storage alongside 6 separate torpedo launchers. Whilst this wasn't enough to halt construction initially it ultimately saw only 11 other ships constructed, though with far more spread out deployments than the Atlas class due to their ability to do the work of 3 Atlas'.
During their time in the spotlight the Proxima class's career was ultimately generally rather eventful. Their distribution saw the ships Split relatively evenly between the Romulan and Klingon borders, supplementing the Atlas class groups. The ships, typically escorted by Miranda class vessels, were often used on flagstop tours around worlds near the Neutral Zone, even diverting to duties such as Second Contact missions when the need arose. One ship, the USS Procyon, was ultimately present at the renewal of the Treaty Of Organia in 2278, where it was found that Organia was completely empty. Upon testing if the ship's weaponry was functional, it was discovered that the Organians had completely ascended away from this plane of existence. This caused relations to break down as Ambassador Kor of the Klingon Empire and Ambassador Carlos Rivera of the Federation came to vocal blows upon this discovery, as it gradually became apparent that the War of 2268 had ended for nothing. It took the personal intervention of Admiral Robert April and the arrival of the 4th fleet, which initially escalated the issues further, which allowed tensions to diffuse for long enough for Federation-Klingon relations to avoid exploding into all-out war.
Across the 2280's the Proximas maintained their position, being moved more towards the Klingon border as it became apparent that the Romulans were too busy rebuilding their fleet to attempt an actual war. 2 ships, the USS Sol and USS Beta Eridani, were notably sent to intercept the USS Reliant when it became apparent both that she wasn't responding but that one of the 3 extant Genesis devices had gone missing, though they were still two days away when the Reliant was destroyed. The two ships were consequently kept nearby at Starbase 32, near what was then the Genesis Planet, when diplomatic tensions got hot. When tensions cooled they returned to the border. During the Whale Probe incident two other ships, the USS Proxima herself and USS Procyon, were diverted to try and intercept the probe, unfortunately running out of fuel before the ships could intercept. Another ship, the Alpha Centauri, diverted to try and assist the USS Constitution after she was affected by the probe, unfortunately arriving as the ship drifted into the Neutral Zone. Unfortunately this saw the ship destroyed with all hands by the Klingon Empire, almost sparking a war due to the nature of what had happened. If it wasn't found that the Constitution had been illegally "testing the waters" with the Neutral Zone, it's likely a war would've actually broken out.
As the 2290's came and went newer battleships in the form of the Excelsior's, Yamato's and Georgiou's soon replaced the Proximas in their starring roles, and so they were gradually rotated to lighter border flag-showing duties. In this role they proved fairly successful. The ships were ultimately refitted with a pair of the same SSWR-XX warp cores as later Miranda class ships, losing their two additional cores in favour of more space for fuel and stores. These refits directly prolonged the Proxima's lifespans, with the ships maintaining their roles on gradually less important borders until their replacement, in the form of the Freedom class, finally replaced them in the 2350's. None of the ships were preserved, though allegedly one ship, the USS Alpha Centauri, can still be seen being used as a storage hulk at Qualor II. Ultimately their legacy is one of a clear product of their time, a better ship than any single Klingon vessel but one with substantial drawbacks. But overall they didn't exactly fail.