While I can’t address why some Celtic or Germanic warriors might have gone into battle shirtless, I can certainly address why defensive armor like mail was relatively rare. Although I’m largely discussing Germanic barbarians during Late Antiquity, most of this applies to barbarians from earlier periods.

While barbarian arms and armor were roughly equivalent qualitatively to their Roman counterparts, quantitatively the difference was immense. This is especially true in regards to defensive armor. Mail corselets were relatively rare, as were iron helmets. Only the wealthiest fighters could have afforded them (local nobles (optimates), cantonal kings (regales) and their armed retainers (their comitatus)), and would have worn them with any regularity. Most barbarian warriors would likely have carried only a shield for defense (usually round and with a boss, perhaps rimmed in iron), carried in tandem with a thrusting spear. In both the archaeological and Roman literary record, other forms of armor outside the shield are uncommon. This was not because the barbarians could not produce defensive armor, but rather because barbarian society lacked the organizational sophistication to manufacture them in vast quantities. During Late Antiquity, Roman arms were mass produced in state-run arsenals throughout the Empire known as fabricae, with the Notitia Dignitatum recording that certain fabricae often specialized in producing different types of arms. These were organized and run by the bureaucracy of the Roman state, which drew upon the wealth of the entire Empire to supply and fund the production of arms. The economies of scale created thus allowed for arms to be produced at a lower cost, meaning that most Roman soldiers throughout this period had mail armor, an iron helmet, a shield, and a spear and/or a sword. In contrast, the relative expensive involved in the small-scale production of such equipment restricted their availability to only the barbarian elite.

The concept of the unarmored Celt comes from Roman and Greek historians, particularly Polybius' writings on the Punic Wars. Diodorus Siculus, a Greek historian, also wrote about the military accoutrements and fighting tactics of the Celts. Tacitus also comes to mind, but the fighting tactics he most vividly described were limited to the Celts of northern Britain, little detail of Celts elsewhere.

Polybius describes Celts as fighting nude, using short, rounded swords unfit for thrusting (Histories 2.33.5). It is this description that many later historians copied. Polybius' description was also widely depicted on monuments, like the Victory Arch at Orange.

Meanwhile, Diodorus paints a more vivid picture of a helmed warrior, sometimes nude, other times clad in mail (Bibliotheca 5.30.2-3). There have been helms and scraps of armor found very much like those he describes, the Waterloo Helmet in the British Museum, and the Satu Mare helmet in Bavaria, so archaeological evidence provides some confirmation of his description. There is also linguistic evidence the Romans borrowed the very concept of mail armor from the Celts, as the term hamata of lorica hamata, the descriptor for mail armor used by the Romans, is not believed to be a latin word.

So the short answer is yes, some did fight nude, but why? Caesar, Strabo, Diodorus, and Pomponius Mela each describe a Celtic belief in immortality and reincarnation. So strong was this belief that even business arrangements could be delayed until after death (Mela, de situ orbis 4.4.4).

Ultimately, yes some Celts did fight without armor, choosing faith over iron. But that was not exclusive, some also wore mail.