In my country (for people with a so called "dynamic pricing" contract) the price of electricity already drops significantly when we have a load of green energy (wind, solar). I guess this is just a side effect of not having a well-balanced grid. But still that's already kind of a CO2-based pricing - although not directly

This is what makes me wonder how much additional benefit an explicit carbon signal actually buys you over normal dynamic pricing

Mostly yes, but it works the other direction as well: a fossil power station may not want to completely shut down because it expects more demand later in the day, and offer a few hours of low cost electricity to stay in the merit order.