There are any number of ways you could do it. Different services would perhaps be best funded in different ways.
For example, fire and police services should be allowed to charge working individuals a flat fee (a napkin calculation suggests this might be $5-$10 a month). The armed forces could be allowed to advertise and seek voluntary contributions, perhaps even going so far as to 'crowdfund' specific projects. Medical treatment could be funded by a pool of contributions (as the British NHS was originally conceived - it worked pretty well). Parents should simply pay for schools. This sort of thing would let individuals see exactly what they're getting. Companies could publish their tax contributions, earning some level of public kudos in return.
I'm not suggesting this as a blueprint for 'how I'd run the government', but the fact remains that income tax is:
1) Unfair. Your income bears little or no relationship to your use of government services. In shittier parts of the world, private companies actually provide public services that the government can't be arsed to deliver ... and then get taxed for doing it.
2) Hard to administer. What exactly is "income"? Tax authorities tie themselves in all kinds of knots trying to decide what income is and which jurisdiction gets to dip their sticky fingers into the pot.
I agree that people who are offering car transport services should make a contribution to government coffers, simply because (a) they're using government-provided roads and (b) they're generating a lot of pollution. You could have a mileage tax, a gasoline tax, or electronic toll points. There are all sorts of options. There is no need to make a distinction between a 'professional' and a 'casual' driver, because in reality there is none.
Uber, as a legal entity, a corporation, doesn't have any obvious obligations to compensate the government for services rendered, because they are (I'm guessing) just a smallish bunch of guys in an office. Their moral obligation to contribute to Society At Large is a different issue.