I am a big fan of the pre bronze-age-collpase civilizations, and something I note is that they LOVED chariots, and makes sense, it gives mobility, some protection, you can have archers in the chariots and you can move around soldiers quickly through the battlefield.
After the bronze age collapse chariots were still popular, but something happened, now infantry seems much more important, and cavalry becomes a thing, there are still chariots, but they are no longer the core of the army, until eventually no one used them anymore.
Finally, when Cesar went to Great Britain he mentions that the people there used chariots, and by the way he describes them it seems he had never seen them in combat, it's almost like if the Celts were behind in the tech tree, but I see no reason for chariots to be an inferior technology
There are a lot of theories about chariot use in the LBA. Some argue that they were primarily tactical: mobile archer platforms, as you say. Others that they were primarily prestige markers, not meant to win battles per se but instead to announce the wealth and status of the owner (and by extension, his faction). I think it is a mix of both, and elements of both contributed to their decline in the 1st millennium BCE.
As tactical weapons, chariots have some serious limitations. They work well on flat, level terrain, but struggle mightily in rocky, broken, and uneven terrain. Compared to the horseman, the chariot is cumbersome, expensive to construct and maintain, and less able to maneuver.
On the other hand, I would argue that there was always more to the chariot than simply a weapon of war. Chariots were incredibly expensive, complicated, mobile billboards of status. The fact that we find so many of them in graves, all across the Near East and Europe, suggests that they should be considered in the same category as other prestige grave goods of the LBA/Early Iron Age: "killed" iron blades; metal cauldrons/tripods; exotic imports like faience beds; gold in all forms; expensive ceramics; and animals (like the killed horses at Lefkandi). The decline in chariots in general might have more to do with changing attitudes towards prestige display (out of graves, away from individual aristocratic display, and towards "useful" display such as dedications in sanctuaries). This is a general trend which we can track across cultures in the early Iron Age.
There is a lot to unpack there (entire theses worth, in fact), but tl;dr: probably both tactics-based, and also related to a shift in cultural norms towards the use and display of wealth as elite status.
You mentioned Caesar: interestingly, Caesar arrives in Gaul/Britain in the middle of the 1st century, precisely when those cultures are entering the Iron Age proper and are undergoing similar shifts in the perception of elite status. I am not an expert on La Tène archaeology, but my general understanding is that we see the same kind of decline in expensive, elite burials and a general shift to votive deposits (of swords, for instance) in sanctuaries. This is the same general process which happened in the Greek Iron Age, in the Italian Iron Age, in Spain, and elsewhere.
There is a great chapter by Robin Archer, "Chariotry to Cavalry: Developments in the Early First Millennium" in New Perspectives on Ancient Warfare, eds. Fagan and Trundle (Brill 2010), which addresses your question in a lot more detail. The argument there is that cavalry naturally evolved from chariots for a number of tactical and cultural reasons, as I suggest above.