A friend of mine runs a successful business bringing over doctors from the Philippines, Pakistan, etc to work in the NHS, so the gap gets filled that way.
This is far from true. On the margin, doctors can work in the private sector or leave the country. Only a few choose to do so, but this is (among other reasons) because the NHS always has to respond to price pressure to keep that small number from growing to where it causes shortages.
Recently a change in how pension contributions are taxed was pushed through almost entirely because it affected senior doctors. If the NHS had pricing power as you suggest, why wouldn't they just tell doctors to pay up?
I know two junior doctors, one is working in Australia and the other is still in the NHS and would never countenance leaving no matter how rational a decision it may appear.