There's a lot of fans of solo diving in this forum area, as you'd expect. It seems to be a passion to champion the practice. That said, I see much of the emphatic support for solo diving arising from a belief that it's "safe" or "safer" - usually on the basis of one person's varying experience diving alone. Nothing's gone wrong for them, so they believe nothing can go wrong. Sometimes that emphatic support arises from a desire to 'legitimise' the activity - and thus further legitimise their activities.
I caution against applying what might be right, or safe, for another person... as automatically being right, or safe, for yourself. We all have unique capabilities. We all have unique experiences to draw upon. We all have unique physical and physiological traits. We all have unique strengths...and most importantly, weaknesses. As such, anyone who says; "Yes, it's safe...do it" is really doing you an injustice - especially if they don't know you, and your unique characteristics intimately. In short, the internet is not the place to seek approval for your activities.
What can go wrong when solo diving? The answer is no more, or less, than can go wrong when buddy diving. Not very drastic huh?
So why do/did the main certification agencies always counsel against this? Well... that comes down to the resolution of problems. Whilst the risk of having a problem is the same, or arguably less, a solo diver will receive zero support to resolve an issue, should it arise. Unless you believe that problems are better resolved alone, that lack of support means that a problem, should it arise, is less survivable.
Personally, I don't know of any human endeavour that is better tackled solitary, rather than by a team. That's just my belief. I don't know of anyone who doesn't benefit from a helping hand when they are under stress or in difficulty. If you are the victim of a problem, then your psychological and/or physical capabilities are deteriorated by the predicament that you are involved in. Another person isn't. They can offer you assistance. Even a little assistance can drastically improve one's chances at breaking out of an incident spiral.
So, the crux of the matter is whether the diver concerned is capable of dealing with any predicament they may find themselves in, without assistance. That's a very individual question - and the answer to it lies in the total culmination of that persons knowledge, experience, training and unique physical/psychological make-up. Naturally, we all want to imagine that we have the capacity to look after ourselves. To do otherwise is to snub our own egos. Our egos can lead to very self-deceptive decisions sometimes.
The solo diver needs to be confident of their abilities - but this confidence shouldn't be driven by a subjective ego-driven self-assessment. It should be based upon evidence. It should also be reinforced by peer/mentor review. The evidence you look for is past performance under worst-case scenarios - both diving and non-diving. That means drawing upon your experiences. How did you handle problems underwater previously? How do you manage your stress under pressure?
If you haven't had sufficient experience to yet encounter serious problems underwater, then you're not in a position to use experience as an assessment tool. Likewise, if you've never been in a highly stressed situation, you have no capacity to judge your psychological thresholds. Sorry, but that's true. In my opinion, 'highly stressed' means you're thinking "I could die now", because you could die now - coupled with a short time-scale to effect a life-saving resolution.
If you can't use experience for self-assessment, then you need to seek an alternative. That alternative is a suitably experienced (see above) peer, mentor or instructor. They will/can put you into safe, simulated scenarios to assess how you would perform. That, to me, is the critical psychological component of solo diver, tech diver, cave diver training. Most divers who go through this training enjoy a shocking realisation about the true level of their capabilities. That's a great catalyst for learning and development. It also tends to turn an over-confident diver into a cautious one.
Safe solo diving (as the OP partially identifies) consists of 3 factors, in this priority;
1) Psychological capability. ("keeping a calm head") As explained already, this is the capacity to self-resolve any foreseeable problem that can arise. No assumptions should be made in this regard, when deciding your suitability for solo diving. Don't let your ego write cheques your capability can't pay.
2) Appropriate reactions. This is the 'tool set' at your disposal. The skills and procedures that you have ingrained to be reliable solutions to foreseeable problems. This is a product of suitable, effective training - followed by practice, repetition and refinement. 'Knowing' a skill isn't enough - appropriate reactions need to be ingrained at an instinctive level. I'd use a Bruce Lee quote to illustrate the difference I am trying to describe; "I fear not the man who has practiced 10000 kicks once, but I fear the man who has practiced one kick 10000 times." Basically, a skill can't be relied upon until it's been practised 10000 times. The diver with 10000 skills, who hasn't ingrained them, doesn't have any skills.
3) Equipment Support. ("a pony bottle") Most novice or inexperienced divers put the most emphasis on this factor, but really it's the least consideration. A safe solo diver has to have the equipment necessary to resolve problems. That equipment is useless without the skills or psychological robustness to utilise them in a real emergency. For the solo diver, this equipment must replicate that which would otherwise be supplied by the existence of a buddy - primarily an alternative, independent air supply. Depending on the nature of the diving, this could include redundant buoyancy, gauges, lighting, navigation, or communications/signalling equipment...even a mask.
Number 3 can be achieved with a credit card.
Number 2 can be achieved with hard work and time.
Number 1 can be improved with training and experience, but is ultimately a unique personality trait.
