Just because you could does not mean you should.
There's a variety of options. The questions are are they better and, if so, how and why?
The proximate reason for the decline in funding and the consequent decline in healthcare provision in the UK is government policy and ideological rigidity. The NHS is being induced to fail as a deliberate act of policy - the minister is a co-author of a book advocating NHS privatisation. There is nothing inherently wrong with the basic mechanism of universal healthcare that is tax funded - to the contrary, it is the only solution that a civilised progressive society would adopt. Tax is not a penalty on' hard working families' it is their investment in an orderly, inclusive, civil society.
Multiplying the funding sources might yield additional revenue but it would definitely add to the complexity and cost.
When seeking to reform a complex system the first step should always be to add simplicity. Applying this principal to the taxation system, by eliminating all scope for avoidance, would yield a much higher level of investment in our society and allow all 'hard working families' to benefit.
Competing interests: No competing interests