Tax funded roads being gated off to those without higher incomes doesn't sound fair to me in any case, even if there are alternatives.
Okay, so as a solution to that, let's stop paying for roads with taxpayer money, and instead let roads in Manhattan be funded entirely from private funds, collected from those who use the roads instead (either individuals who don't mind paying, or businesses who pay fees on the deliveries, the same way ships pay fees at port).
That way, taxpayer funds can go entirely to public transit instead, so (assuming we solve the problem of rampant corruption and fraud within the MTA and NYS government[0]) it's a win-win for all!
I'm being half-facetious here, but those are the implications of the argument you're making, and I'm not going to say they're necessarily a bad idea.
[0] https://www.nytimes.com/2017/12/28/nyregion/new-york-subway-...
Edit: Elsewhere in this thread, someone posted a link to Beijing's solution to congestion, which seem interesting:
https://en.m.wikipedia.org/wiki/Road_space_rationing_in_Beij...
Unfortunately, I don't know if I would trust the politics of NY to allow for that.
From what I've heard well off families in Beijing will then just have multiple cars, one with an odd license plate and one with an even one.
That said, especially given the circumstances of Manhattan, congestion charging probably makes the most sense and also has the virtue of bringing in additional money that can theoretically be used to improve public transit.
[ADDED: I think the key here is that driving in Manhattan for individuals is something of a luxury good. We'd view things differently if this was about distributing relief food after a natural disaster.]
Lotteries are an absolutely terrible system. They fail the basic test of Pareto efficiency. Or in other words: lotteries result in outcomes in which one person could be made better off without making anybody else worse off. The only way to avoid this with a lottery is provide a secondary market, and that defeats the whole purpose of a lottery - at that point, you might as well skip the lottery and use a direct market.
Not all Pareto-efficient scenarios are good, but all scenarios that are not Pareto-efficient are bad, by definition. If Alice could be made better off without making Bob or Eve (or anyone else) any worse off, we know there's a problem.
Say there's a national park campground that fills up within a few minutes of opening up reservations.
However the lottery is implemented exactly (wardialing, picking applications at random), it's pretty much how you have to do things. Raising prices until supply meets demand is (properly) politically a non-starter even if it's the "correct" economic solution.
You can have a waiting list (as I believe many Grand Canyon permits use) but creating multi-year waiting lists have their own set of problems because it makes it hard for people to plan.
But, yes, in general the alternatives for excess demand of a fixed quantity item tend to be:
1.) Increase price
2.) Have a lottery (which doesn't really make any sense here)
3.) Have some sort of rationing scheme (waitlist, can only buy X of item, etc.)