The tax would claw back less than 20% in your 'print billions for the NHS' model.
I don't see how you reach that conclusion as every spending transaction carried out with the injected money will attract taxation in some form or another. The only time that it won't yield a return in tax is if it removed from the economy by being 'saved'. And 'saved' in a mattress under the bed at that, because most savings attract interest or dividends which are taxed.
And, your 'model' seems to be based on a constant outflow of money to the NHS with no return. Which is illogical; there is a 'return' in the tax take which is then reinvested in the NHS. The tax take is also higher because there is more money around to be taxed.
Try tracing the progress of £100 through the economy. As tax removes part of it every time it's used for a transaction the amount available for 'use' becomes smaller and smaller until it is eventually all, or nearly all (let's not forget savings) returned to the government as tax.
Although MMT has differences from it a good deal of it is classic Keynes. To call it a 'theory' is also a misnomer. It is a description of how money 'works'. And, of course, Richard Murphy is not its only exponent, he's just easy to read on the topic. There a plenty of reputable economists who support it.
It's also worth remembering that economics is not a science, it has a strong sociological element to it. How people behave with money has to be factored into economists' theory.
P.S QE did cause inflation; it caused inflation in the investment markets; which, of course, have very little connection with the 'real', day to day economy.