Quite a fundamental question. A follow up question I'd love to have explained if the answer is no: then in exactly what forms can electric charge that's causing electricity in a solid conductor, be made of? Is it the ions in a battery?
I think the answer is no.
Electricity is fundamentally the motion of charges and there is no process to 'generate' charges, the best one can do is reallocate charges.
One thing which is possible is that you can have net charge zero in neutral conductors, typically electric wires.
So to say electricity without charge is not sensible but it is possible to have zero net charge.
Fundamentally, an electric charge creates an electric field that repels like charges and attracts opposite charges, which is intuitive. However, a positive charge creates an electric field that points away from that charge. A negative charge creates an electric field that points toward that charge. BUUT the charge only moves if there is a NET electric field acting on it. When you add a battery to a circuit, the electric field is not 0 and flows from the positive side to the negative side. Electrons move in the opposite direction of the electric field to try to balance out the electric field vectors (think of speed vectors). Also, the electrons would be moving opposite the passive connection (plus to minus in a resistor for example), but the direction doesn't really matter as long as your consistent. That's my best answer. If you think I stated something wrong, let me know. Learning relies on feedback after all.
I clearly think that the answer is yes. Take, for example, a case of a capacitor. We know that DC does not pass through it and AC does. Well, if electricity was the flow of charges, then theoretically AC shouldn't pass through the capacitor. Then, why does AC pass through it? The answer is simple. We notice that there is one difference in case of an AC and DC through a capacitor. In case of AC, there is a varying electric field which frequently changes its direction with frequency of AC. This becomes the core idea for why AC passes through a capacitor. You might have already understood the reason. Yes, like changing magnetic fields produce electric fields, changing electric fields produce magnetic fields and thereby current. This is the famous Displacement current that Maxwell introduced.Clearly, this is not the flow of charges,right?
For more information, you can refer NCERT books and Halliday, Resnick, Walker's books.Thank you.