I'm not super invested in this debate as a whole, but I do feel the need to chime in on this. Having played both titles fairly extensively back in the day, GoW3's combat has nowhere near as much depth as DMC3. It's important to note that there's a significant difference between simply having a truckload of moves that you can look cool with and having a significant degree of depth to your combat system, and while GoW3 (and the entire first GoW trilogy) has the former, it pales in comparison to DMC3 on the latter.
Combos in GoW3 that basically infinitely stunlock most enemies are incredibly easy to perform even on Chaos, and while you can go out of your way to look impressive by switching weapons on the fly and throwing in different attacks, there's no real gameplay incentive to do so; you won't be doing more damage and the game doesn't really have any other method to encourage moveset exploration beyond that. As long as you're familiar enough with enemy attack patterns to know when to drop your current combo and get out of the way of an incoming attack, you have the knowledge you need to beat the game, even on Chaos.
DMC3, on the other hand, has a much greater degree of depth because it has, at the risk of sound pretentious, let's call them "high level techniques," that are almost essential for the player to familiarize themselves with if they want to make it through the higher difficulties (Very Hard/DMD) without having to spam healing items. If you're not, at the very least, familiar with using different styles, jump canceling, and DTEs, you're going to have an extremely difficult time getting past even the first third of the game. There are advanced strategies that the player essentially needs to learn if they want to survive higher difficulties. On top of that, the game's inclusion of a style meter, influenced by the variety of attacks a player uses, not only provides a constant incentive to use different weapons, but also factors into how much in game currency you earn at the end of a level, adding an additional incentive on top of that.
Devil May Cry 3 handles its gameplay depth better than God of War 3 because of the things you stated. Doesn't make it lots deeper. Its just that Devil May cry 3 has better AI and enemy waves and the style meter is amazing incentive to mix it up. Still God of War allows nearly the same skill ceiling just doesn't require you to be that good to beat the game.
I would say it does require lots of skills to do the Challenge of the Gods in God of War 3. In the campaign, Chaos mode requires you to be really good but not to the extent that Dante Must Die in Devil May Cry 3 does. Still the option is there for those who need that depth. Watch the videos I linked there. The developers didn't know such combos were possible and that shows the amazing depth it has.
I'd argue that's exactly what depth is. Perhaps this dissolves into a debate on semantics of what words are, but to me, depth in an action game means that not only does the game have complex mechanics, but the game also requires a high level of skill to properly use implement them in battle. And that, in turn, requires at least a high enough degree of enemy AI competency to make certain situations ones where it would be proper to implement those mechanics. The issue with GoW 3's depth, imo, is that there are no real situations, at least in the campaigns of 1-3, where it's ever proper to implement extended, flashier combos with weapon switching because there's no point where it's more effective than simply using basic combos with the Blades of Exile (or whatever weapon you've put the most upgrades into). I would argue that a game has to provide situations where the player is forced to think intelligently about what they're using and how in order to really have depth.
I think another viable comparison is fighting games, which share a lot of similarities with action titles. Games like Marvel vs. Capcom 2, for instance, have a lot more possible moves and combo variety than, say, Street Fighter or Mortal Kombat. Yet I don't believe you would find many people who claim that Marvel 2 has more depth than comparable games of the latter series; in fact, people usually argue the other way around. Why? Because having a sizable number of moves doesn't mean a game has depth. That can certainly be part of it, but it isn't a guarantee, or even the main factor. Despite the sheer size of Marvel's movesets, there are usually only two or three combos that high level players consistently rely on; basically any situation where you land a hit will prompt the player to use the combo that will deal the most damage (if they think they can kill) or deal the most damage/place the opponent in a positional disadvantage (if they can't kill). At the end of the day, those are the only two optimal strategies encouraged by the game.
And it's the same deal when it comes to GoW 3, there's no real incentive provided by the game to do anything beyond just sticking to whatever weapon you've got the most upgrade points in. If you're switching weapons, you're almost assuredly doing it for coolness' sake. Which isn't a bad motivation, but it means that the combat isn't particularly deep. It also doesn't help that 3 of the 4 obtainable weapons function very similarly, thus negating a lot of potential depth that could have existed had there been a bit more variety. Contrasting this with DMC3, where not only do all the weapons function very differently, but the game's superior AI and enemy design means that you are required to think about how you're going to approach different situations and choose different strategies based on what you're fighting. That, to me, is depth.
I did watch the combos, just to check if people had discovered a much greater degree of technical skill than when I last checked, and, no, they haven't. I think that speaks more to the devs thinking their combat was more limited than it is; which isn't a bad thing, many times devs find out that something they didn't intend to be in the game is in the game and it actually improves things a lot (Super Metroid, imo, is one of the greatest games ever in no small part to many of the sequence breaking abilities Samus has that the devs didn't intentionally put in the game).