Customer acquisition cost per new customer is the quick version of CAC: campaign costs divided by new customers won, plus a check against the contribution margin of the first order. For a quick assessment of a campaign or channel, that is enough.
The central question is always the same: is the customer acquisition cost below the contribution margin of the first order? If yes, the campaign was profitable immediately. If no, the customer lifetime value must carry the difference — and that value then needs to be realistically achievable.
For strategic decisions (channel selection, budget allocation) this metric is not enough — that requires the full CAC with repeat purchase rate and customer lifetime value. For a quick check on the success of a single campaign, it is the right tool.