Absolutely, on all points. I'd perhaps add one other - we've got potentially good diagnostics, but they're not used for this, they're rare and they're horribly expensive. (Problem is, it's longer and less clear than yours.)
An example. Hospital MRI scanners are around 2 to 2.5 T, which gives sufficient resolution to see severe injuries and malformations but not much more. Medical scanners can go up to 7.3 T and research scanners actively used go up to 9.1 T. At this upper end, blurred sections of the brain are almost crystal clear. You can see not quite to the neuron level but fairly close. Subtle issues can be detected. It's more than good enough to find out if there's a problem with mirror cells, bandwidth issues (too much or too little) and similar fine-scale deformities. The best scanner that can be built that can take a human head is around 13 T. It's unclear what this would show, I've not been able to find any info on it
I wouldn't ask psychiatrists and neurosurgeons to have an underground bunker with dozens of such devices armed with top technicians at the ready, although if one of them is the sole winner of the US Powerball at its current 1.2 billion dollar level, it would be nice if some of it was spent on such things. However, MRI as a diagnostic tool is strongly discouraged, apparently, which seems to defeat its value as a means of rapidly identifying and classifying evidence you can't otherwise get to.
I counted the total number of scanning technologies (excluding minor variants) and came up with 33 different diagnostic tools that could be used at the level of the brain. Of those, I have only known two of those to be used in practice (EEG and MRI), never even remotely close to the levels of sensitivity needed to analyze the problem unless, as I said, there's a problem at the grossest of levels. EEG, for example, is performed with as few leads as possible and the digital outputs I've seen look like the ADC is cheap and low resolution. Nor have I ever been impressed by the shielding used in the rooms (the brain is not a strong source, so external signals matter a lot). I've read papers where MEG is used, but it seems to be almost exclusively research with very, very few hospitals actually using it.
This doesn't contradict your statement that there are no good diagnostic tools, partly because nobody has the faintest idea if these tools would be any good in diagnostics (as it's forbidden by the great overlords), how you'd read the data (if it's not actually used then nobody can understand the output at all, and if it's used but never for diagnostics in mental illness then there's no means of understanding what the output means in this context).
That's just the bog standard medical gear, though. Whilst it should be useful (your experience shapes your brain, your brain shapes your experience and this recursion should mean that you can identify traits of one from the other), there will be other tests. In fact, there are. There are hundreds of questions that make up the official test for autism spectrum disorders, but I've only heard of (and then second-hand) one doctor actually running through them. Most glance at the DSM (which is worse than useless and the criteria listed in it are largely rejected by both the checklist and those definitely in this category) and that's it. The checklist is probably not optimal and is probably incorrect much of the time as autism has a very wide range of causes (both known and suspected) and congealed categories of unrelated conditions won't work with any single checklist. Researchers hotly dispute even when it can be diagnosed and at what age it first appears. That's clearly not very helpful.
But that's positively enlightened compared to something like "Borderline Personality Disorder" (a label given to anyone who doesn't fit any billable category and is generally considered not worth wasting time on by the medical and psychiatric professions). Here, there really isn't an actual diagnosis as such, just an identification that there's a problem and that it's not something insurance will pay for.
We need a good, solid ontology of mechanisms of mental illness that forgets tradition and costing, whether there's any lawful or technological way to detect them or not, because those can be measured (even if only in principle) consistently and reliably. It's not enough, by a long way, but at least there will then be a clear indication of what the gaps are, what precisely we don't have diagnostic tools for (and perhaps even theory for).
This would also be the starting point from which medicines or therapies can be developed. You're right about side-effects. About half my current medications are there simply to counteract the side-effects of other medications. One I was put on, temporarily, shut down my colour vision. The problem? Doctors had to experiment on me. They had no idea what would happen until they tried me on something. I really do not like being used as a lab rat when the long-term effects of even short-term exposure is unknown but where it's known that even the short-term effects include death even at the lowest end of the therapeutic range with no understanding of why or to whom this will happen. It strikes me as... all a bit vague.