I would love to see hypothecation, esp with respect to environmental taxation as a transparent market correction for missed “polluter pays” costs.
I think trying to compare private medical insurance premia with personal contributions to NHS is a bad exercise and its going to need some clear wording to explain why they are not comparable.
Private domestic waste collection though could be one of those services that you ought to be able to compare cost and service with the public sector option.
EDIT: BTW it really would be a miniscule cost to implement. The calculation is formulaic based on the tax calculations that HMRC already have to do for your tax statement. Theyre already posting it to you, so there’s no additional distribution cost. Maybe one extra page of material. Costs almost zip.