You don't seem to understand the concept of externalities at all.
>If we taxed based on negative externalities and rewarded based on positive ones oil companies would own literally everything.
Negative externalities are costs that you inflict onto other people without compensating them. Positive externalities are benefits third parties receive without paying for them.
This means we already reward oil companies by letting them have free stuff and letting them monetize that free stuff. This is at the expense of all the people who weren't born into free stuff. We live in a world where "Oil companies already own literally everything" to a large extent.
If you have monopoly rights to a positive externality as a third party (e.g. extraction rights to oil), you get to charge for the full surplus, essentially making it so that it doesn't matter whether the positive externality exists or not for the rest of the economy. Privatizing a public externality as a third party is as simple as selling it.
Taxing CO2 cuts into the monopoly surplus and leaves less profits for oil companies. That is the defining feature of pigovian taxes. If we assume a reduction in income taxes, then the dead weight loss of income taxation shrinks and we get less income inequality.
>If we rewarded based on positive ones oil companies would own literally everything.
The producer of the positive externality of oil is a collaboration between plants accumulating carbon and the earth's crust compressing it into oil. You could also call the producer of this externality "the environment".
This means that if we wanted to reward the environment for its good positive externality producing behaviour, we should stop destroying it globally.
You appear to be under the misconception that the third party benefiting from the positive externality also produced the externality. E.g the oil company gets an income from oil, therefore it also produced the oil via photosynthesis and compression according to the neoclassical productivity hypothesis. This is not case by definition, rewarding the production of positive externalities would not reward oil companies, because they didn't produce the positive externality, they just extracted it.