Well, I have never trained in a Koryu art, but I have heard some people describe their experience. It doesn't seem as regimented in the sense that there is more personalized instruction, and students sometimes learn material in different orders, or are shown different ways and variations on a particular technique. As far as rank, I was only aware of students being awarded a certificate if/when they were allowed to teach. That, among other things, seems in contrast with more modern Japanese Martial Arts. But, that is only my perspective.
Okay… the first thing to realise is that Koryu is a way of categorising a particular grouping of martial arts (well, not just martial arts… but that's getting a bit complicated…

based on the age and cultural histories… in other words, a Koryu art is a Japanese art… not a Chinese one, not a Korean one, not a Filipino one, not a Western one… and traces its' founding to prior to the Meiji Restoration (1868)… although that is also slightly up for contention…
What you might note there is that there is no other defined criteria (there are certain hallmarks and traits that are found, but nothing set in stone, so to speak) for what is or is not a Koryu art. There is no single form… no one teaching pedagogy… no single approach to ranking, or lack of ranking… no single requirements or authority to teach that is applied across all Koryu… to hear a few people describe their Koryu experience is just that… a few people's Koryu experience. So, while I wouldn't necessarily disagree with the impressions of the people you've spoken to (or, a little more realistically, your impression of what they told you), I would caution against taking such descriptions as definitive or equally applicable across the board.
For example, while it is very common for Koryu to be taught in small groups, and therefore to offer much more personal and direct instruction, which can lead to a fair amount of personalisation of the way the material is presented, it will still follow largely the same procedure for everyone. Sure, you might learn the kata of the Shoden section in a slightly different order to another student, but that could be due to your personal understanding or affinity for certain aspects… but you still won't move onto the Chuden section until you've gotten a real hold on the Shoden. As far as being less regimented… well… that'll depend on a range of factors… it can be both less regimented and more so at the same time, really. Rank is also very much up to the individual school… the most common is the use of Menkyo (licences), indicating to what level your education in the art has progressed… but some arts don't use them much, or use a highly abbreviated version… or use a more modern dan-I ranking system… or use both… the Menkyo ranking can include a ranking for teaching… or it might be a separate licence entirely… it might be only able to be attained at a certain point, possibly even not until you've attained the highest "regular" rank, Menkyo Kaiden (in most cases… not all…

, or it might be something you can get almost straight away, depending on circumstances and location… or you might have to attain a mid-level licence first… or anything else.
When you mention that these approaches seem to contrast with modern Japanese arts, not really… I can see how you can get that impression, but really, it's largely superficial. The application of ranking is different, but serves the same purpose in many ways… the pedagogy of teaching, when it comes down to it, is also largely the same, just on different scales… they really do match pretty well.
But, to be honest, I wasn't thinking all that much about the words I used. I was mainly thinking of Okinawan Karate, and then just lumped it in with Japanese Martial Arts
Ah. I think about the words I use a fair bit…
Okay, the next part I'm going to break up… there's a lot to cover...
One thing I am curious about, and would like to ask you, is about the nature of the Koryu arts we have today.
Okay. The first thing I'll say is to reiterate that you really can't generalise Koryu as if they're all the same, or even the same "type" of system…
As I understand it, they focus on preserving the art --
Well… yes… but then again… no…
Overall, yeah, it's a characteristic of the approach of Koryu… but it's not really the only focus… and is really more a part of the current mentality of most of them. It's a lot of shades of grey, really.
one that, you might say, has been "dead" for a number of centuries --
Uh… no. That is one thing I'd never say… mainly as it's completely not true or correct.
by which I mean it is not applied, and it does not change.
Who says?
What I'm saying is that those ideas are not actually supported by the ryu-ha themselves, or their practitioners… on either count.
One of the biggest contrasts I see with CMA is that CMA, on the other hand, tends to change a lot with each generation, and leads to a wide diversity of styles.
From a Koryu perspective, that would be the same as founding a new martial art each generation… which happened in some systems… but isn't really the same thing as a continuous lineage. That said, if the CMA change with each generation, how are they considered "traditional" arts? Or arts from history? And, if they're not, then why the culturally defined movements that don't necessarily fit the cultures they've been imported into (both in terms of society/geography, and time period)?
So I wonder, at what point were what we today term Koryu living arts?
At all points. They've never stopped being "living arts".
I am aware that many lineages descended from others, so there was surely the same kind of innovation and liveliness about Japanese Martial Arts at one point in time; albiet probably still more conservative in line with Japanese culture.
The thing you need to understand about the various ryu-ha is that they are founded (and propagated) for a range of reasons… in some cases, it's a matter of innovation of combative methodologies… in others, it's to form a repository of military knowledge… so you need to look at each system in turn in order to get to any beginning answer to this question… honestly, it's a big one, and a fair bit off-topic for this thread. Feel free to start a thread in the Koryu section, and I'll see about fleshing these responses out.