Your problem seems to be either "what do I say to prospective employers asking why I'm leaving so soon?" or "how do I avoid this in the future?"
Prepare a short summary indicating the direction of travel in your career; leave out hint of resentment or blame, no matter how strongly you feel it or how justified it is. Your having been wronged is something you can't prove and the prospective employer probably doesn't care about. You won't get a job because the new employer feels sorry for you, the point of asking why you're leaving is a) to fish for flags like inability to get on with colleagues or manager, being sacked for disciplinary/capability reasons, etc., and b) to check whether your expectations of the new job meet what they're going to want you to do.
You could say something like
My role at the moment is a mix of senior work and technician work, but I'm finding I do well at the senior work and am looking for a more leadership-focussed role.
I was interested in the role of a senior with team leadership elements. As it turns out the roles here were not quite what I'd expected, so I'm looking for something more challenging that better uses my skills.
Of course, that assumes your prospective job isn't also 50% technician work.
Now, you ask
are promises made during hiring binding, or is this situation the employees' fault for not being cautious enough?
Well, yes, actual promises made on behalf of the employer in hiring are generally considered binding in the UK, but they're worthless unless you're willing and able to enforce them. Specifically, you say:
promising that their position would become a team leader position after the first few months, with an appropriate pay raise
Promising? Unconditionally? In writing? Even though verbal agreements tend to be legally binding in theory, generally, unless it forms part of the written offer, it's going to be really hard for you to enforce that (and it isn't easy even if the wording is clear and written). It's sadly not uncommon for companies to do this, for instance because the hiring manager thinks at the time the company is committed to a particular path. Collectively, maybe, if you have a union and the company keeps making and breaking promises, you might be able to hold them to account for it.
So yes, I would say if you accept a job based on something that was not written into your contract then you are not being sufficiently cautious. If employers do make an offer having mentioned something verbally that would make the difference between you accepting an offer or not, ask for it to be in put in writing in the offer.