There's no such thing as "road tax", VED isn't hypothecated nor are fuel taxes. A proportion of income tax, stamp duty, inheritance tax, VAT and other taxes a cyclist might pay go toward road maintenance. A well paid cyclist is probably contributing more money towards roads than a poorly paid motorist, but nobody argues that gives them any more entitlement to use the road. Likewise the taxes involved in using a motor vehicle don't give motorists any more right to the road than cyclists. It's a public highway maintained out of the common public purse for public benefit.
The common argument (used by the van driver quoted in the article and in some of the comments) that using a motor vehicle involves paying more tax than using a bicycle so gives a greater entitlement to the road really winds me up, both as a cyclist and from a political ideology.