Not exactly. Unemployment insurance, if run privatly, would probably be more akin to household or medical insurance; with premiums based, in part, on the liklihood of unemployment arising, so varying with ecconomic activity and the persons occupation, and the payouts possibly being related to the premiums (as with pensions). The payments would also be made from the assets of the fund held by the insuring company, meaning they could be lost and no payments be made at all (unemployment insurance would seem especially vulnerable here, as claims are likely to increase in econonomic downturns, just as bond and stock values decline meaning less assets are availabel to pay out). With a government backed NI scheme government spends the money as it is collected, then uses money collected at the time fo claim to pay out, rasied from taxation or borrowing. The strings are a little more stringent than that (at least in theory) with evidence of what efforts have been made to find work needing to be shown to claim.

Having the unemployed do some work might be helpful for them, as it would (in the case of long term) demonstrate to a potential employer that the candidate has the frame of mind to attend work and be punctual etc. It is almost certain however that even the most carefully thought out plan of "forced work" of the kind usually mentioned, street cleaning for example, or taking up unskilled or manual work with private companies, would be at a cost to jobs of people already working in such areas. Nearly all private companies and even councils would want to take advantage of government subsidised temporary labour rather than hire their own staff at the market rate.