The problem is that this isn't ideological, caused by rapacious capitalists at the top trying to bleed the system dry. I've mentioned this before, but I think it's a good example of the sort of challenge the NHS faces in its current form.
An employed registrar (for example) will earn around £50,000. However, a locum brought in temporarily will earn more than twice that on a daily rate.
So over the last few years there seems to be a steady production line of newly qualified doctors not going into permanent roles, creating thousands of positions that need to be temporarily filled. And coincidentally enough, there are thousands of newly qualified doctors who will happily be paid £500 per day, to do the same job as the guy (not) doing it for £50k per year. Whack the whole lot through your "Dr. Smith Medical Services" limited company and pay 20% tax on the whole lot as well instead of higher rate of income tax.
No blame on the doctors - it's their labour and expertise, and if you don't need the stability why wouldn't you do it the way that pays you twice as much and gives you complete flexibility about how much you work? It's not like they are taking much of a risk on whether the demand for their service is there. No blame on the trusts for paying it either - their choice is pay through the nose for staff or have (more) people dying.
That's the reality of "paying private agencies for medical staff". But I don't really see what the fix is - just insisting that they "employ actual NHS staff" doesn't really deal with the root of the issue.