The Honest Version
The buses are fine. The island is just much bigger than them.
Naxos's public buses are cheap, clean and punctual, and on the corridor between Chora and the west coast resorts they run often enough to be genuinely useful. If your holiday is Agios Prokopios in the morning and Chora in the evening, you do not need anything else.
The trouble is scale. Naxos is roughly the size of Malta, with a mountainous interior, a road network of well over 200 kilometres and around forty inhabited villages. The bus serves perhaps a dozen destinations. Beyond Filoti and Apeiranthos the timetable drops to two or three departures a day, the last one leaves the mountains in the late afternoon, and whole regions — Melanes, Alyko, the entire east coast, most of the Tragaea — have no scheduled service at all.
That has a knock-on effect people underestimate. Without a vehicle you cannot have dinner in a mountain village, you cannot be at Alyko for the sunset, you cannot follow the wind to a different beach when the meltemi picks up, and you cannot go to a panigyri. The bus decides your day; you just attend it.
Taxis exist and are reasonable for short hops, but there are not many of them, they cannot be flagged down outside Chora, and the return leg from a remote beach is the one that gets expensive.