It's rarer to just find someone looking for just a "general electronics engineer" ie I need schematics, PCB... That doesn't mean these roles don't exist, but they may be more oversubscribed and would likely be done more by perm employees. You'll probably do better in a niche field, one where it's not worth hiring someone to work full time. Especially because you can command a higher rate.
I agree. None of my regular customers came to me wanting a 'general' electronic engineer; they all wanted an engineer with specific, proven experience in their own particular product area - or at least, something close enough that it's obvious that the skills and requirements are transferrable.
Tip: don't advertise yourself in terms of your electronics skills. Instead, advertise the kinds of industries and products you've worked on, and/or the type you want to work on in future. Most of my customers don't have an electronics design dept of their own, and don't especially care what types of components or circuits I use to solve their problems.
What I deliver is a solution to a problem, usually consisting of CAD data, a BoM, firmware, documentation, and support. Nobody really cares about the implementation details until much later - often when it's already in production and they just want to better understand how everything works (and knowing already that it does, in fact, work).
From what you've described of your skill set, my honest impression is that contracting might be hard work for you, but I'm speaking from a UK position, as a perm employee, so YMMV.
What leaps out at me, is that you're academically qualified and have 14 years' experience, yet you describe your own skill level as 'junior' in terms of anything complex, and the tools you're still using are hobby grade at best.
Why is that? What have you been doing for those 14 years? Designing, or just soldering?
Being a consulting engineer (and maybe that's not what you mean by 'contractor'...?) means you need to be able to confidently walk in to a new customer's office, listen to their needs, and using your own skill and experience, decipher what they really mean, and convince them that you can offer a solution. Or walk away if you know you can't.
Don't underestimate how hard this is. The people you meet won't be electronic engineers, they won't speak in the same jargon and won't necessarily frame their requirements in terms of an electrical spec. Remember that they're hiring
you to be the electronics expert.
The first part of any job is to come up with an engineering specification, ie. a list of deliverables that actually will solve your customer's problem - including the parts of the problem that they don't know about yet, but still expect you to take care of. Remember again: there's a good chance they're looking for
you to make their electronics problem go away, and that's the
whole problem, not just the bits they're explicitly spelling out on day one.
Their needs could be tightly specified, which is great when it happens. They could be trivial, to the point where you feel bad even charging for your time. They could be vague. They could be impossible. They could be vastly more difficult and costly than they're expecting. Worst of all is when they're implicit, and neither of you even mentions them until it's too late and you get the "doesn't it even do...?" phone call.
My advice? Try and get a full time job in a design capacity, where you can develop your skills and, if at all possible, meet some customers.
Be aware, some companies like to build a wall between customers and engineers, as though allowing the two to meet is somehow dangerous.
It's not. Customers are just people, trying to make a living, and they have problems you can help them with. 99% of the time they'll be delighted to see you; the other 1% is when something has gone wrong
and they're not getting the support they expect.
Just don't be
both of the reasons for that 1%.