Would the Gods Approve of Neo-Pagans?

In my exploration of modern pagan paths, I have found two main types of path: neo-pagan, and re-constructionist pagan.

Neo-paganism is any path that invents its own rites instead of trying to reconstruct ancient ones, while reconstructionist paganism is a path that tries to reconstruct ancient rites.

People of reconstructionist paths often dislike neo-pagan paths, because they feel people of these paths do not worship the gods correctly or think of them the right way (I saw an article in which an Asatru person bashed Wiccan pantheism).

Recently, I started reading the book "American Gods" by Neil Gaiman. In it, Odin disapproves of neo-pagan paths, accusing them of being "pretenders" with inadequate worship. (He wants more ancient rituals, I've heard, but don't quote me, I haven't gotten to that part of the book yet.)

Even though I'm not a theist, I like to keep my mythologies and characterizations of the gods consistent. If reconstructionists don't feel that the gods would approve of neo-pagan paths, but neo-pagans do, are these people worshiping the same gods? Might it be more accurate to say that they are worshiping two different gods with two different characterizations? Are Wiccans worshiping the "Wiccan" aspects of the gods?

I prefer not to think so, because many neo-pagans take a more pious attitude than reconstructionists will give them credit for, saying things like 'Apollo will not let you forget he is a god'.

I like to characterize the gods as follows: They prefer ancient rites, but as long as reconstructionist paganism is a minority religion, they are willing to settle with anything, since every bit of thought or belief helps. (This is based on the "clap your hands if you believe" trope, which says that gods become stronger or weaker based on how many people believe in or think of them). I also characterize the gods as drawing power from their domains, so "inadequate levels of belief" can be replaced by participation in the domain (you could serve Thalia, the muse of comedy, by being funny).