The tories would love it, though, as people who see money coming out of their income to contribute to the NHS would complain even more about the poor service they are paying for...
'Complain' is an ambiguous word though. If you mean 'show dissatisfaction so that the NHS can be more easily replaced' it is one thing, but if you mean 'are dissatisfied so the government has to make improvements or risk losing support' it is another.
The NHS, as we are always being told, belongs to all of us. People 'complaining' because they can't see a GP, or because they have to wait months or years for operations can, or should, be a force for change. If the public remain passive it is far more likely that the government will continue to spend less and reduce the service further.
Speaking out against cuts is neither disloyal to the NHS nor somehow playing into the hands of the Tories. Also, if income tax does not fund spending, what would happen if it were abolished? We saw the economy crash after the Truss/Kwarteng tax-cutting budget - that doesn't suggest that taxation doesn't matter.
Not only that, but (rightly or wrongly), people are told that the reason most of us pay income tax is to fund things like Health and Education. Whether that is true, done so that there is less resistance to it, or something else, that is what most people believe. So yes, if we see a huge chink of our earned income go on tax and don't feel we are getting what we are supposed to be paying for, we will 'complain'.