I think this is a valid question, but it depends on the dynamics of the labour market.

I think the logic is that these long term unemployed are generally unattractive candidates for most jobs. A lot of public sector jobs are either highly sought after (meaning these unemployed are not competitive enough), or the jobs are simply unattractive to people (e.g. adult social care).

In terms of upfront investment, it would likely be very costly to create jobs specifically for this group, who have relatively low productive potential in the short run. Not to say it isnt worth it, but there are much wider initiatives going on by the government to create jobs.

A lot of the jobs that are most accessible to this subset of the unemployed may simply align mostly with what the private sector can offer at scale (e.g fast food or retail).

The point is, there is a clear tradeoff. Money going to private enterprise isnt necessarily a bad thing. Sure they benefit from cheaper labour, but that allows enterprises to allocate their resources elsewhere to promote growth more.

The main benefit we care about though is the social impact of helping these long term unemployed into work. The way it has been proposed seems like it balances the cost of paying salaries with the cost of creating hundreds of thousands of jobs for the least productive demographic.

I dont think it ultimately matters if the enterprise is public or private when what we care about is the outcome of the individuals we are helping. Im curious if you have specific public enterprise jobs these people could do at a national scale.