Thanks again everyone for your thoughts on my original question. I think there’s a lot here that can help ANY testers wrestling with similar issues about relevance.
(Even though my original question was self-centred - sorry!)
Also interested to hear people mention cases where an apparent demand for automation skills turned out to be not quite as it seemed. Yeah, I’m sure there is an element of “fashion” to automation’s appeal - just like every org now has to be Agile without really knowing what that entails.
A lot of you have rightly talked about educating and advocating within our organisations on the value of testing etc. This is definitely all good advice. I agree, of course.
Indeed I’ve done that … admittedly without as much success as I would have liked …with my current team by introducing them to ideas beyond their sets of test cases, like Session-Based Test Management, cognitive biases etc.
For me, unfortunately, a big issue is that I need to leave my current org and find work nearer to home (and stop the 3hrs+ a day I spend commuting) for family reasons - and for my own sanity. 
There aren’t many testing jobs near me anyway, and when something does comes up it’s typically focused on automation/coding skills. I struggle to remember the last time I saw a job ad that made any reference to exploratory testing being a desirable aspect of a candidate - most of them don’t even make any reference to thinking skills.
And a testing role that doesn’t value exploratory testing isn’t appealing to me.
I know that those kind of roles do exist, in the “right”" kind of companies, but they’re not common. I truly wish more (all?) software hiring managers thought like Michael Bolton but they just don’t.
(See also the community’s emphasis on blogging, twitter-presence and being at the right conferences - as per James Bach’s “build your reputation” stuff. My blog isn’t great but I do emphasise its existence on my CV etc to show hiring managers that I don’t just do testing, I also think about it. But I’ve had more than one interview where it’s become clear that they haven’t looked at my blog or twitter and have no particular interest in them.)
Before testing I was in marketing and there you needed to remember, unless you happened to produce products aimed at people in marketing, the customer was not you. They didn’t live like you, didn’t think like you, didn’t like the same things as you.
Similarly I’m finding that many people hiring testing resource aren’t like me and don’t see testing in the CDT-influenced way that got me enthusiastic about it.
It does seem that ability to automate is now perceived as a requirement for a tester to some degree.
Which is fine - it happens in different kinds of industries all the time that people need to re-skill as the world evolves. But it’s not something I can sell myself on with my minimal coding skills. And after years of self-learning in my own time (I started with no technical background at all) on top of a long commute I’ve come to the conclusion that I now have better things to do than taking part in the endless programming/tools skills arms-race.
And then, yeah, even if the skillset isn’t an issue, there’s potentially an age one. Chris described it well. I suppose until you reach a certain age you don’t even think about how many job ads contain (perhaps innocent) coded references like “we’re a young, fun team”. In 1 or 2 cases where I’ve been told that I was clearly a strong candidate but didn’t get jobs because of “fit” I always wonder whether that’s a legally acceptable way of saying I’m too old. (The truth probably is that I just don’t want to admit I smell bad and am super-annoying.)
Anyway, I don’t want to seem like I’m wallowing in self-pity here. I know that if I’m not thriving in testing that’s my own responsibility and I could make different choices.
I’m sure testing as a skill has a future and its right that those involved in testing should constantly learn and adapt. The ideas and comments you guys have contributed here will help testers do that.
I just feel that I personally don’t really fit what most orgs are looking for in a testing resource. So my future probably lies elsewhere.
Not sure where exactly, but that’s a different problem. 
In the meantime I am off to enjoy the Easter weekend and as much chocolate as I can stomach!