Yes, that's right. The spot price is determined by the marginal source of electricity. That's normally gas, but is sometimes coal, imports, or pumped hydro storage. The consumer actually benefits twice - once when the wind is sold into the market, which necessarily lowers the spot price for _all_ electricity (more wind supply is the same as reducing total demand, which will reduce the market price). Then, the difference between the spot price and the CfD price is refunded, but this time only on the electricity actually generated by wind.
In general, electricity suppliers don't literally pass the spot price on to consumers. It used to be that a supplier could price electricity however they liked, including passing the spot price on. However, for the past few years, suppliers are required to buy futures, and price their supplied electricity on that basis, because of the domestic price cap. If they didn't they would run the risk of going bankrupt like Bulb. Futures bring stability, and in an efficient market with perfect information buying futures would be the same as buying spot. But the market is not efficient and with perfect information, so that assumption doesn't apply. Notwithstanding that, you were correct, the part of electricity generation attributable to wind should be taken account of at the lower price.
For most consumers, their marginal cost of electricity is identical to their average cost (ignoring the standing charge) as they are on a fixed rate tariff. It would undoubtedly be more efficient to charge consumers the actual marginal cost at the time of consumption, and then refund the CfD payments in a monthly payment later, but that's not how it's done sadly.