In practice, the real reason the Ancients claim not to interfere with the 'mortal' realm is because the authors of Stargate realized that the Ancients should just be able to fix all the problems (except the Ori) with a wave of their hands, and that wouldn't make for a very interesting story.
That said, in-universe the evidence strongly supports the Ancients reasoning being a combination of near-fanatical hatred of the Ori and anything to do with them, and the fact that the majority of the Ancients really were assholes. This actually curves around back to my 'super-powerful children' comment; the Ancients generally behave in an incredibly childish manner, refusing to take responsibility for their fuckups, refusing to even admit they fucked up in the first place, getting angry at other people for breaking their rules despite having no way to have known of those rules in the first place, doing incredibly stupid shit (half-ascended Anubis) in a fit of pique without any consideration at all for the consequences and more. These are all the actions of a petulant child, and the thing about children is they haven't properly developed empathy yet; children do not instinctively\emotionally understand that other people are also real people too, a fault that many of the Ancients show repeatedly throughout the series.
The Ancients would rather fuck over half the galaxy in order to 'prove their point' and 'teach a lesson' to one of their own than actually address the problem. They're more concerned with getting their own way and putting others in their place than anything else, and it shows.
The Nox at least are willing to help some of the time, and most importantly the Nox don't leave their planet-destroying toys lying around where anyone can start fucking with them. In comparison to the Ancients, the Nox are quite responsible. Which only serves to highlight just how irresponsibly shitty the Ancients are in comparison, because outside of a few specific instances the Nox are totally okay with just sitting back and watching the Goa'uld genocide and enslave entire societies.
Also, if you go through the series and count all the times that the Ancients do fuck about with mortals, it becomes blatantly obvious that their 'we do not interfere with the mortal realm' bullshit is just that; bullshit. They don't interfere with the mortal realm, except when they do.
Incidentally, most of the time when they do they're just making things worse by either lashing out at someone who 'broke their rules' or taking stuff away from the mortals, usually after said stuff has already caused an only-just-barely-averted catastrophe, and just as the mortals finish cleaning up the mess and sit down to start figuring out the thingy and learning some SCIENCE! the Ancients show up to take their ball and go home, because the mortals aren't 'ready' or some nonsense.
Seriously, the Ancients are an entire culture of nigh-omnipotent, petulant children, a state of affairs that has only been exaggerated by their existence as glowy energy beings insulated from the consequences of their actions. The number of Ancients that demonstrate a visibly adult attitude can be counted on one hand, and they're all individuals who have had regular interaction with mortals and thus have experienced the consequences of their actions; a necessary step in the process of growing up.
The Asgard are the only decent non-human aliens in the entire Stargate universe; not only do they have a legitimate reason why they're not swooping in and fixing everyone's shit but when everything finally goes to pot for them they actually gather up all their tech into a big ole database and give it to the SGC. Neither the Ancients nor the Nox would have been willing to do similar in the same situation, no-one is getting their technology without prying it from their cold dead hands.
e: Their claim that they are not gods is also pretty laughable if you think about it, if you went to any non-Abrahamic human and described the Ancients: Energy beings of great power that can shape reality itself to their whims and gain even more power from the worship of mortals, and then asked that human what they would call such a being, they're going to look at you funny and go 'sounds like a god to me. Duh.' Because that is totally what ancient humans thought gods were. (And indeed, is totally what modern polytheistic humans think gods are.)
They are basically the living definition of small 'g' gods; not an omnipotent creator entity, but beings of overwhelming personal power who draw strength from worship. If they didn't draw strength from worship then I could see an argument against godhood, but once you're a metaphysical being of power who draws strength from worship, you are a god, whether you want to be or not. Of course most of that can be put down to cultural trauma from the Ori, but it's interesting that the Ancients apparently decided that because the Ori were abusing their power, the Ancients should therefore deny their power entirely rather than use it responsibly. Maybe if they ignore it for long enough, it will go away? (Oh look, more childish reasoning.)
