Technological solutions already exist. Creative solutions are called for.
User fees need not be difficult to administer!

For example, the EZ pass electronic transponder system for autos could be
easily be extended to small planes. Aircraft owners would be required to
pay a small annual fee for the transponder, say $10,000. As you pass by
the OMNIs, charges to your credit card could be automatically posted.
During takeoffs and landings, the same transponder detection equipment
could be utilized to charge. Perhaps a first missed approach would be on
the house. For subsequent missed approaches, a 50% landing fee would be
charged.

Your radios could also be equipped with electronic debiting software, to
charge the card in the event you request flight following or need to
contact ATC. Newer planes could be factory equiped with instrumentation
(like the Hobbs) that would show how much you're racking up on the AMEX
card. If you reach your charge limit while aloft, a fuel shut off switch
could be automatically engaged, thereby encouraging timely payment of the
user fees. If you are at sufficient altitude, there should be time to
contact AMEX to get the credit limit lifted in order to accomplish an
runway landing.