Well, duh ..
By extending the rules to the private sector, not only would the government raise some much-needed cash, but it could also stem the exodus of contractors to the private sector. In the medium-to-long term, at least.
When it does, freelancers will have nowhere to turn.
Oh yes they do: abroad. They'll join the other exodus from Blighty, namely the people who actually do the work in health care.
Is it just me, or is the long term intention of the UK government to end up with a country that appears to be the landing zone of the B Ark? Freelancers don't follow this strategy as a path to escape the clutches of HMRC, they do it to actually have a moderately decent wage left as taxation has reached frankly ridiculous levels compared to abroad.
If HMRC was really intent on collecting taxes they'd go after the people who have made evasion into an art and companies who sell services here but don't pay any tax on them but I guess that's too much like hard work. Robbing the peasants is easier.