Do middle-aged non-ninja developers belong?

I think as long as you can code (you said you're not a natural, but I trust you're still at least average at it) and you have a desire to push yourself (which does come across), then there's no reason to stop coding. After all, who wants to conform to trends ;)

My main bit of advice (and something which I'm sure doesn't need to be said), is make sure you have a back up plan in case your passion doesn't translate into a profession

I am ferociously dedicated - to an almost quixotic level - to programming, but in general I find that I literally have no compulsion to do any coding for myself outside of work hours.

I'm almost in my mid-thirties and I took worry about the future where the need for codehorses like myself is thoroughly outweighed by the need for "Rock Star Ninjas".

Sure, there's the manager strategy, where you'll trade one set of concerns for another, and that's probably an easy way out. You might also consider something like a 'business analyst', or a position which sits between developers and non-developers, translating non-technical requirements into specifications for developers to work on.

In short, there are options for 'very technical' people (which developers tend to be) to transition into less technical areas of the same field, and they are necessary jobs as well. You might talk to your manager, boss or (if you have access) a CTO to see what kinds of opportunities are available. You might even try 'interning' for a week or so in those various positions if you can be allowed to and see how they actually interest you.

Regardless, programming might be a young man's game, but there are a myriad of things that I bring to the table that it's hard for youngsters to match. I am proficient in a variety of languages, frameworks, etc. I can build an application from the ground up. I can administer the database, I can build the frontend with HTML and Javascript, I can use Backbone, Ember or Angular to make it super interactive, and I can of course build the backend in Django, Python, .Net, Java, etc., etc. The years of experience you have make you ideally suited for speccing things out too - it's often easier for someone who's worked with a lot of systems to look at a given pair of systems and see where things overlap, where they might fit together, and what an integration effort might look like.

There's a job for this too, called IT Architect. Again, you might find that you love it, or you might hate it, but it's a real thing that demands technicality and expertise.

Since you have a steady and mostly happy job, focus on differentiating yourself after hours. Build out your github profile (my current focus since it's entirely private projects right now), find a programming niche that is financially rewarding and put your emphasis there. Specialist programmers in under-served sectors can earn significantly more than the management supervising them. You don't get to that point by "doing .NET" or building WordPress plugins (not to sound condescending, I've done both for a living), but it can be incredibly lucrative.

It may seem that "rock star ninja developers" are a young person's game and to some extent it is. Many people find that their life priorities change as they age. Many younger developers may value making a name for themselves and devote a lot of time outside of 40 hours a week to coding and upping their game. So even if younger developers may have less on the job experience, they may be far more active in the development community than career developers. Those factors may be just as or even more important to some companies than tenure. Other companies may place more value in stability and tenure than having a bunch of rock stars. As you age and your life priorities shift (such as having kids with lots of extracurricular activities that you want to focus on), the type of position that suits you best may change.

To grossly oversimplify, startups will gravitate towards rock stars - cheaper and and flashier with a lot of drive, while enterprises will go for reliable, tenured developers who consistently get the job done even if they are just average. I've seen no shortage of middle ages .net and Java developers. Might not be the sexiest languages, but they are very popular with enterprises. Any company should be interested in a developer that is an accomplished learner and stable developer. But if your priorities in life don't include spending a significant amount of non-work hours diving into the hot language/technology du jour, you won't be able to compete with younger developers who can.

So yes, there is a job market for average middle-aged developers, it's the same job market for all developers. But if you're suited to an enterprise position you'll have a hard time finding one that lives on the bleeding edge of technology of which you might be wanting to get into.