What we need is a cryptocurrency that is somehow mined by pulling carbon out of the atmosphere.
There's a lot of great work on the "this is a huge problem, how do we fix it?" side of things; we need to keep encouraging more on the "there's an abundance of this stuff that people consider bad and worthless, how can I use it?" side of things.
Imagine if every CO2 molecule were somehow unique: carrying with it a physical hash that could cheaply identify the processes that produced it. In this scenario, such an economy would become feasible.
To reduce complexity and cost, it needn't be every molecule. Say just one in a quadrillion, at an exactly fixed rate.
Of course CO2 is actually generic, as far as I know. Yet with only one wave of the magic wand, an economic solution has appeared.
Assuming regulatory and societal support, which should be forthcoming in this current climate (pun intended).
The beauty of proof of work is the hashing algo is both 1) very hard to solve and 2) very easy to verify by anyone anywhere.
Furthermore, 3) miners have a financial incentive to solve the algo and 4) They may do so in anyway they see fit.
At a minimum a crypto-like solution should have these four properties.
Now we already have a lot of unpaid CO2 miners (plants) doing it inefficiently, ad hoc and short-term. The first stage should be to allow these to opt in to get rewarded. Like you could mine bitcoin with spare CPU back in the day.
Most mining devices don't actually ever successfully win a single bitcoin. But they get paid pro rata for their contribution to the combined hashing power, which did.
For miners in a pool hashing power is the saleable commodity. Perhaps this was the point you were making.
Even if it didn't fulfill its potential (yet), we must remember what made bitcoin so special: eliminating the double-spend and central authority problems. How do you intend to sell your sequestered CO2 and prove that it's still yours to sell?
This is all hypothetical anyway. To my knowledge there's no scheme that satisfies those criteria wrt proof of sequestration.
You could ban oil as a store, but people will use it as a power source for net positive emissions processes to sequester CO2 in another form.
It also makes money laundering laughably easy (plant trees). Then again, maybe we care less if we can turn that money towards good.
And there are also the general issues with using a physical currency. It's hard to transact across distances. You could do a bank-like system, but at that point it starts to look like carbon credits and calls into question the necessity of a parallel currency.