I've read about Hallstatt and La Tene cultures, but they didn't cover nearly the entire area of the Celtic-speaking world. What other ethnicities existed that primarily spoke Celtic? Why was the language group so wide-spread?
To be clear: Celts did exist. The terminology of Celtae actually derives from a self-identifier used among some Celtic language speaking peoples in antiquity. Though it would not necessarily accurately applied across multiple societies, the term Celts can quite reasonably be used to refer to Celtic language populations identified as such by Greek and Roman accounts during antiquity. It wouldn't be correct to describe all Celtic language speaking populations as Celts which, along with the fact that the term Celtic also derives from Celtae, can make this distinction somewhat confusing.
This is further complicated, as you note, by the identification of archaeological sites as 'Celtic' in a material cultural sense. These concepts are something of a leftover from earlier scholarship which directly related sites of this sort to not only Celtic-speaking populations generally, but as direct ancestral groups to the later Celts made famous by Caesar. The modern picture is not nearly so simplistic, and so this usage has generally fallen from favor (particularly in Archeology), although it has not entirely been discarded since many of such sites have been shown to be legitimately associated with Celtic-speaking populations, and also there still remains some credible demonstration of common material, social, and religious cultural themes across them.
The most direct answer to your question regarding the overlap between the archaeological conception of Celtic (La Tène and Hallstatt as you mention) and the linguistic conception, is that they simply are operating under different definitions and parameters of 'Celtic'. The categorization of 'Celtic' art and archeology covers a far smaller geographic range, that was simply a partial representation of a vast spread of language, throughout which there was not always, or even often, also shared cultural aspects that would appear in an archaeological record.
As to why this language family spread so far, the simplest answer is we don't really know, although there is plenty of theorizing! The original paradigm of Celtic language spread nestled the linguistic 'homeland' roughly in the central part of the suspected maximal extent, and vaguely plotted it outwards from that point, with Hallstatt and La Tène as chronological snapshots of some of that development. More recently a new hypothesis, that Celtic language arose in the Atlantic Zone, and spread throughout the continent from there, has been proposed. Barry Cunliffe and John Koch's multi-volume Celtic from the West being the vanguard of that theory. This subject has been discussed recently and that thread may be of further interest:
Why the languages spread so far is not known, and is probably impossible to know, though their dissemination across the broad expanse of the Atlantic coast is substantially more demonstrable if they arose in that region. Trade and settlement along coastal regions in European antiquity being far more active and far-reaching than the inland equivalent. It has been hypothesized that Celtic language may have served in this context as a lingua franca of trade, and the commonality of shared religious elements between otherwise distinct Celtic-speaking cultures (the Druids being the foremost example) may also indicate a potential for some religious aspects to also have been represented in this linguistic transmission. This subject is presented extensively in Celtic from the West although those works are not intended for a general audience, so would not necessarily be an advisable introduction.
The best further reading on the subject would likely be Barry Cunliffe's The Ancient Celts (1997) which is an introductory work which thoroughly explains the differences and developments of the various concepts of Celtic and the relationship between them.
