Solo diving, is diving without assistance. A newly qualified diver needs assistance, regardless of their confidence level. Confidence can exist in the early stages of a diver's experience simply because they have not yet been confronted with problems. It is different from the confidence that is gained from experience and the knowledge that you have been exposed to problems and learnt how to deal with them.
Every scuba agency espouses the 'buddy diving' system, for good reason. Whilst divers have a freedom of choice, they should be aware that their decision making and the possible negative consequences of this have a direct impact on their families, friends and even the rest of the scuba community.
I would never say that a diver cannot go solo diving. But I do recommend that divers take advice from a suitably qualified and experienced mentor or instructor about their capacity to solo dive safely...and then follow that advice.
I disagree that all newly qualified divers need assistance but that's a different debate, I believe.
Confidence of how to deal with emergencies is, in my experience, the same no matter how that confidence is gained. Understanding how and when to react is different than knowing the exact steps of what to do. If a person doesn't know the exact steps of what to do, they should go back and take their OW class again. Knowing the timing is entirely different. Understanding the difference between these two concepts is also a big factor in being "prepared" to solo.
I am of the opinion that everyone coming out of OW should (and I know I do) know how to respond appropriately to any underwater emergency. Free flows, OOA, underweight/bouyant ascent, overweight, blown o-rings, entanglement, loss of vis or mask, loss of other equipment, or an animal "encounter", equipment/gauge failure, currents. Is there anything else that isn't similar to one of these that I might have forgotten?
Barring true medical emergencies, the possible variations of SCUBA emergencies fall into a few basic categories and can be dealt with appropriately by knowing how to deal with them categorically. Every single one of these types of problems were covered in my OW class. How to deal with them generally and specifically. Being able to actually function in those kinds of emergencies isn't something OW classes have the time to train. It's up to the individual to practice these and learn more, on their own or in further classes.
Does that make someone safe to solo? That depends upon who you ask and, in my opinion, the conditions they are going to solo in on a given dive. A pool is certainly different than Puget Sound.
I'm confident I can deal with any emergency during a solo dive. At the same time, I believe I don't have the appropriate diving skills to dive solo without causing some of those emergencies (I'm still working on ascent rate, and buoyancy for example) and I don't have the redundant equipment I feel is a requirement. I have practiced buoyancy and checked out gear in a pool "solo", though. Risk/reward, cost/benefit analysis.
Like you said, the key is being aware of the possibilities and ramifications when making your decision to solo.