I may be misinterpreting, but it seems to me there are two separate (?) things going on here.
I'm sure there's more than 2 separate things going on here. But I'll settle for two as it sounds much better
I generally interpret the term "action" as connoting something more specific than "approach". You seem to have the opposite take ("action" being more general than "approach"). So that's a little confusing.
Maybe not as much when you understand the context that in previous discussions I've suggested approaches before that were shot down due to not being specific enough.
But aside from that, you seem to be saying that there should be a roll anytime there is uncertainty about the outcome of whatever the PC does as opposed to uncertainty about whether it achieves the stated goal. The problem that I see with that is that I am pretty sure one can always come up with something that is uncertain about the results produced by an action. So that doesn't seem helpful.
I think the following are the basic game states to do with actions and goals:
You succeed at your goal and face no setback
You succeed at your goal and face a setback
You fail at your goal and face no additional setback
You fail at your goal and face an additional setback
If checks can only determine whether you succeed or fail at your goal then there is no success with a setback . It can't exist. I firmly believe success with a setback is a great tool. I bring this up because it seems to me that your criticism also applies the same to it.
That said, I don't believe it's a valid criticism for what I'm describing. Why? Because while the check is there to resolve uncertainty, we only need to resolve "meaningful uncertainty". I think there's 2 tests that must be met before something is considered to be "meaningful uncertainty". The uncertainity must rise to a sufficient level - so something like a 1/1,000,000 chance of something happening isn't meaningful. The uncertainty must also be of that nature that it's fun to resolve or important or interesting to the scene or story. This will always be a subjective call based on what you are looking to provide in the play experience and what your players expect to gain from it. For example, some tables may find it totally uninteresting and unimportant to roll to see if you can help your ally up the stairs after being drunk. Others may have a great deal off fun in that scenario. If they do then it's meaningful uncertainity. If they don't then it is not.
I think that's the way around your criticism, but if for some reason you think it isn't, can you explain how that same criticism doesn't apply to success with a setback and if it doesn't how it gets around it?
A secondary comment is that you have positioned this as "alternative" to goal-and-approach. As has been noted several times goal-and-approach is a description of a style for players to communicate to the DM. You seem to be explaining an adjudication approach. Not the same thing.
As used on this forum by many posters it's much more than a description style for players to communicate with the DM. It encompasses that as well as an adjudication approach.
In this second part you've described in particular an approach to setting DCs. I think you have a really good point. It's something that I think I've done on an irregular basis without realizing that I was doing something different, so it's good to have it explicated.
Thank you.
However, I don't agree that this really contradicts what the PH says about stealth because I don't think the description in the PH is intended to be limiting or comprehensive. The PH describes a simple procedure for simple situations; it doesn't say that procedure covers all situations. I understand that in other editions (and some other games) rules are assumed to be comprehensive unless otherwise stated. In 5e, it seems to me it is the other way around.
Sure, it may not be stated that it's the only way to do it, so I can't really argue that RAW forbids this thing. What I can say is that it doesn't actively encourage it and a cursory read through of the PHB can easily lead to the belief that stealth is handled by comparing to passive perception in most every situation.
So if it leaves open the possibility for more then that's great! It should have actively encouraged handling stealth that way except in a 1v1 pc vs npc situation because as we have seen in this thread, there's still many handling stealth without the understanding of how to set DC's for it - and so they base it almost entirely off the passive perception value. (and some even use multiple rolls to resolve the situation... ick!)