Overall, I can't agree I'm afraid, Andie.
You're right, of course, in that we should all remember what a school exists to do, teachers and non-teachers (*washes mouth out*) alike. However, I invite you to consider this analogy.
Schools, businesses, hospitals, charities, governments and militaries are all organisations (I don't consider the terms 'organisation' and 'business' to be synonymous). Let's say we have an organisation that, for the sake of argument, is a business. Say it makes furniture (like a cut-down version of DFS, perhaps). That's its primary purpose and what it's there to do. There's no point in it being there if it's not doing that. Also, for the sake of simplicity, let's say it has one factory with one production line (naturally it would be more complicated than that, but this is just a simplification of the process). If the factory workers aren't making the furniture, the business is producing nothing and serving no-one so it may as well shut down. But let's assume that we're not at the point of saying the company is folding. The production of furniture comes under the category of 'service delivery'.
As an organisation (any kind of organisation), it has to keep itself running in order to have that factory, that production line and those workers there at all. A number of important functions and processes go into this: leadership, administration, public relations, personnel, finance, IT, etc. (IT is part of these because it is a core resources that nigh-on everybody has to go through to get anything done these days). There are also a number of additional functions that facilitate the service delivery, such as sales and marketing to raise public awareness of the existence of the company's products/services (otherwise the company would have no customers, or in a school's case, no pupils/parents) - this is slightly different from PR in general because it's more specific.
So let's imagine that you have a nice young lady - Sonya - who works in HR for the company. She might be, say, a team leader (i.e. 'lower-middle' management within that department). Now... you go and try to tell Sonya that all she's there to do is support the factory workers in making the furniture, and she'll tell you where to go, because that's not the be-all-and-end-all of her job at all. She and her department function 'at the level of the organisation', in that they provide a service that enables an organisation composed of human beings to function together collectively as a cohesive whole. They provide their service across the entirety of the organisation, intertwined with all other organisational processes. This is because all the departments contain employees - people - so they will all encounter, and make use of, HR's services at some point. The people carrying out the service delivery element are just one group of these. Next (and this goes out to all those schools whose NMs are still under the Head of ICT), you try telling Sonya that she's to be line-managed by the factory foreman, and she'll quit. In short order. Because it doesn't work like that.
Her work and the factory foreman's work are entirely unrelated. Yes, he and his department are one of her clients - a principal client in fact - but that's all. When things are run properly, service delivery is not at the heart of the organisation. There is no 'heart', no single unwavering focus at the expense of all others, that said others all merely serve. All departments are recognised and treated equally in their contribution to the company's success.
End of analogy.
The same principle applies to IT: no modern organisation of any considerable size (and schools are, generally fitting quite neatly within the industry bracket of 'medium business', at least if it's a secondary school) can function without it, and it permeates absolutely everything. Even the caretakers and the kitchen staff have computers at my school (and yes, they do use them in their work). Even subjects like Art and PE make fairly heavy use of IT resources in many schools. Administration (including a good proportion of clerical duties) is practically paralysed if the whole network goes down, as is finance. Personnel, too, stores employee records and salary adjustments, etc., on SIMS or something else IT-related, or it links in (over the Internet) to a centrally-managed system at the LA's end.
The problem this thread was meant to address is one of perception. When dealing with perceptions, terminology and use of language are important. Teachers are used to having assistants in various roles to support their work directly, known variously as 'teaching assistants' or 'technicians' (for certain subjects requiring additional technical support or expertise). You get Art technicians, Science technicians, D&T technicians and so forth. As I said in my OP, there was a time when a school's IT facilities were primarily used as a learning resource; the technical staff were thereby quite fairly classed as supporting teaching. Teachers are used to the idea that when we speak of 'support', be it 'support staff', 'supporting teachers' or whatever else, what this essentially means is people who are subsidiary to teachers, whose roles exist merely to assist in service delivery - i.e. just one element of the organisation. Within the context of education, I feel that that's what the word 'support' is generally taken to mean. We really mustn't blame them for this - that's still the case for many kinds of support staff, and in fairness that was also how it used to be with IT.
However... this has changed!
The change has been gradual, as schools slowly, over many years, came to rely on their IT services more completely (à la businesses, and frankly everyone else out there by now), so it's not really been noticed. It is up to us to inform and feed back to senior school leaders and senior government decision-makers, including related sector bodies (i.e. BCS) and training companies (e.g. QA) the nature of these changes, as we are likely to be the only group that will perceive them directly over the course of time.
Nowadays, the IT resources permeate much further into the core of the school's processes - into pretty much all of them, in fact, including those entirely unrelated to delivering T&L. The ones that continue for that entire quarter of the year when there is no T&L going on. This change has occurred largely over the course of the latter half of the roughly 15 years that IT has been around in schools in its modern (or at least semi-modern) form.
I feel it might be useful to compare ourselves with school Business Managers at this stage: they also function 'at the level of the organisation', providing the core service of finance plus, varyingly, a number of other very high-level organisational management functions that are absolutely essential to keeping the school as an organisation running (which must be the case before we can even speak of T&L). They are also generally classed as Senior Management, which, in fairness, I can see they would need to be more than we would. But either way, their position was created, recognised and appropriately paid (generally on par with various similar-ish roles in industry) because a group of people, or perhaps many disparate groups, realised the need for them and the value of them. We have seen, then, that it's possible to influence the education sector in this way, as it's happened before with the rise of the Business Manager. Now is the time for the rise of the Network Manager, and their technical staff.
So in summary, Andie, I think you're right in that supporting teachers and pupils is a very important part of our role and it should never be forgotten or put down. Equally, however (and this is the problem), it is only part of our role. Also, as you say, it's easier to swallow the pill if you're working in a place that does value and respect you, and tells you so (I can relate, as mine does too). Unfortunately, however, too many of our colleagues are not in this position, and I feel it is precisely because our role is seen as merely supporting others that we don't get the recognition that, as a branch of our profession, we do deserve.
The pool of talent and dedication on this forum is immense. We ought to at least be able to make some headway if we really try. This is also why I wanted to hear from @Dos_Box, as he already knows what it is to achieve change, in indirectly bringing us all together.