if you own a car, don't you use the road more than those who don't? If instead of COE, ballot is use, then car won't be only for the more well off.
On the first sentence which states the a car-owner uses the road more than those who don't - just for the record, this is a slightly different point from "people who don't own cars don't enjoy the roads at all, and thus this is unfair since the roads are funded by taxes and everyone should be able to enjoy the roads". I therefore presume that you are not pursuing that line any longer. In that case, I'm fine. The first sentence may not be entirely true and there are situations which do not match that statement... But ok, let's look at your second sentence.
You suggest that a ballot system would be fairer to you than a COE system. I presume that this is because everyone gets an equal chance to enjoy the roads (through chance) rather than what you perceive as a scheme benefiting the "well-off". On the surface, sure. If you assume that everyone's life is entirely identical and their demands are uniform.
Let's have A and B, and assume that they have identical wages and expenditure. Both can afford a COE. Under the COE scheme, both would have bought cars.
Now let's have your proposed ballot system, and add a few more details to A and B. Let's just say that the ballot allows 50% of applicants have the right to buy a car. A has 5 children in primary school and lives in Jurong East. His workplace is in Simei and he starts work at 8 am. B is a single bachelor who lives at Simei, and works at Simei. He also starts work at 8 am and can walk to work.
If only one of the two can win the ballot here, and A wins, B doesn't - that seems fair. If B wins, and A doesn't? It hardly seems fair to me then. You have merely replaced "more well off" in your post above for the ballot scheme with "luckier at balloting".
Please note that I'm not saying that the COE system is better, if A could not afford a COE and B could, then the same imperfection occurs. My point here is really that any system to control the ownership of cars will have certain rigidities built into the methodology of deciding who gets to own the cars. So let's not fling our hands in the air and proclaim an alternative scheme a clear winner too quickly. Cheers.