Are the Gods *just* anthropomorphic animist projections on natural things? Is Helios literally the sun, Nyx literally night, because their names in Greek are also the everyday words for the thing?
No, and there’s a lot of evidence backing up this answer.
The Greeks had a name for this phenomenon: synthema. The physical sun is a synthema of Helios. Wine is a synthema of Dionysus. Synthemata are not the things themselves, even though we might call them by the names of the things they point to. Synthema is a hard to translate word, it can mean symbol, sacred sign, even “motto” - a guiding phrase or sentence that acts as a simplified version of a higher goal - can be translated as synthema.
But back to the question... One is literally the thing while the deity has that thing as being sacred to them. Apollo is associated with healing and medicine and light but he is neither literally medicine nor literally healing nor literally light. The sun is sacred to Helios and falls under his sphere. He was traditionally prayed to for many things, not because people literally believed he was the sun but had that as being sacred to him. Archetypes and literalism are modern concepts.
Reducing the gods to single, simple, literal things is denying an awful lot of written cultus, mythology, and theology, not to mention gnosis, praxis, and mystery. Gaia is the Earth *and* the soil *and* a goddess with personality and relationships and likes and dislikes *and* a deep spiritual mystery *and* more. If the seeming paradox of that bothers you, Hellenic polytheism might not be a good place to be.
When it comes to names of the gods they tend to often be functional descriptive but can apply in many ways. Plato talks of numerous meanings behind the name of Apollon for instance that connect to his domain. A god whose divine function is connected to the sun makes sense his name would mean the sun. Apollons name means destroyer, washer, shooter and mover of bodies in harmony...these are all things that are connected to his domain. Plato's Cratylus discusses names and the meanings.
This is an incredible rabbit hole to go down, too, because these kinds etymological associations were all over Greek lit. The Greeks were marinating in wordplay, puns, side-rhymes, and played fast and loose with what those associations might or might not mean on deeper levels. Like... The word "psephos" bears a resemblance to "Persephone". The words probably weren't related in a linguistic sense, but "psephos" was the pebble or pot shard or token used to vote to make a judgment, so the word psephos came to also mean judgment, and Persephone aids Haides in the judgment of souls, taking it from a single kingly decision to a multiple person vote. There's stuff like this all over the place, and in the original language all this stuff colors and gives depth in ways that we lose out on in translation. There is no such thing as one-to-one simple literal translation of ancient Greek to English because there's all this allusion via wordplay that they had that gives so many sacred writings double, triple, or more meanings.
If the ancient Greeks saw Helios as the literal sun itself, how does the myth of Phaethon even work?
In the myth of Phaethon, Helios's chariot is taken on a wild ride *without Helios in it* and people on the Earth we're amazed to see the sun (also the word Helios) moving too quickly across the sky when it should still be night time, and when the chariot dipped too low the Earth was scorched and when it flew too high the land was chilled, etc. So the Sun was doing crazy things while Helios was still getting to his stable and finding out the boy had stolen his dad's car for a joyride. In the Greek version, this myth is way more fun to tell and listen to because of the Helios/Helios words make the listener do a little "wait, which Helios, the sun or the god?" but it's all clear by the context. It's also possible to tell the story using only the words "theos" and "harma atos" and "heniokhos" if you're talking to a young child who won't be able to follow the action if you indulge in the wordplay.
But using mythology to try to argue theology in Hellenic polytheism is notoriously fraught, so also consider that any oaths taken by torchlight at night or indoors are said to be witnessed by Helios (the god) even though helios (the sun) is not present. If Helios were literally just the sun, having oaths witnessed by Helios would have to be done outdoors during the day, which would be really simple to do if the ancients thought of Helios that way, but obviously they didn’t.