Do you see a lot of governments setting aside funds to build temples for Hawkeye?

Honestly this comparison is a little silly. On the one hand, it is impossible to prove that anyone believes anything; I assume, but I could never prove, that you believe the sky is blue. On the other hand, the fact that the Greeks not only tell us they believe in their gods, but also show their devotion through the allocation of enormous personal and communal resources, makes it very difficult to argue that they were just really invested in some cool stories and didn't genuinely think the gods were a meaningful presence in the world.

There are countless ways to refute the idea that Greek gods were just like modern superheroes, but I guess the most obvious one is just how much evidence we have that people thought the gods could help them personally (or their community), and that it was possible to make a direct request for their help or advice. Vast quantities of dedications at temples like that of Asklepios at Epidauros explicitly tell us that people prayed to the gods for help and offered gifts in thanks if their prayers were answered. Countless stories around the consultation of oracles show that the Greeks felt strongly that the ultimate authority in moral and practical matters was the gods themselves, whose opinions you could go and ask for at Dodona or Delphi or Miletos or an number of other oracular sanctuaries. The Greeks smilarly used divine favour as a guide in all their endeavours by making constant sacrifices to the gods and reading omens and signs to confirm and reconfirm that their chosen course of action was the correct one. In the wake of either disasters or oracular pronouncements, whole citizen communities would take serious thought about how they might serve the gods better, since this was considered essential to the wellbeing of everyone involved.

It's significant that not all of these forms of belief and devotion necessarily involve a specific deity. I've written before about the Greek habit of referring to "the god" in the abstract, which was intended to stress not only that the will of the divine was unknowable to mortals, but also that divine forces worked on the world whether we could identify them or not. Some of these forces were not even associated with an Olympian god, but were gradually personified as new deities: for example the 4th century BC saw the rise of Agathe Tyche, "Good Luck", as a divinity with her own cults and festivals. This is pretty categorically incompatible with the idea of the gods as a discrete set of superheroes with specific abilities, whose stories you tell because it's cool to envision their specific achievements. To the Greeks, the gods were everywhere, working in countless ways to rearrange human lives and fates as they saw fit. None of this can coexist with the idea that the gods aren't really "real" and don't exist in the physical world we inhabit.

All Greek religious expression is predicated on the belief that the gods really exist and have power to affect the world around us in the present day. They were variously worshipped by different groups of people to suit their needs and to build communities of worship. The fact that these same deities are also involved in myths and cool stories is almost incidental; people who sacrificed to Zeus of Oaths or Athena of the City wouldn't necessarily even have been reminded of those stories as they performed essential rituals for the preservation of their own world.

To my mind, the problem at the root of a question like this is the way that we flatten and oversimplify Greek religion in modern pop culture. Most of our attention is focused on (often distorted and modernised versions of) myth. Maybe we'll learn something about temples. The impression we get is that Greek religion starts with a set of fun stories, and that everything else is just a consequence of how great those stories are.

This impression is not only historically inaccurate (worship of the gods will have predated the emergence of the myths we know by many centuries), but also totally unrepresentative. Myths were not the foundation, or even the core, of Greek religion. Actual ancient Greek religious behaviour is infinitely varied and complex. Even the briefest rundown that I've just given you is simplified in the extreme, doesn't account for local variation or the behaviour of different groups in society, doesn't touch on contemporary thought about the existence of the gods or the inclusion of new gods, and so on. With the sheer amount of evidence that survives, it is doing the Greeks an enormous disservice to pretend that their belief in the gods is nothing more serious or consequential than the enjoyment we get out of stories about Iron Man.