1) The net externalities are measurably negative
2) The coal producers should be paying them
There are clear and massive positive externalities to fossil fuel, and anyone arguing otherwise needs to take a deep breath. They are the foundation of modern civilisation, and we all get non-economic benefits from the era of unprecedented peace and prosperity that we live in. If we price in all the externalities, fossil fuels probably do deserve a subsidy. The renewables debate is about whether we have an alternative that is better than the historically magical substances we call fossil fuels.
Since it has a higher EROEI than solar, if we are going to start pricing in externalities I would be comfortable arguing fossil fuels deserves a higher subsidy than solar for positive externalities. I don't think that is sensible, but neither do I accept that there is an 'implicit subsidy'.
The 2nd assumption is a political question that afaik has been answered in the negative. It doesn't make sense to claim that they 'have a subsidy' when the system we use to determine if they should be paying for an externality (currently) accepts that they do not have to.