The main "weakness" is that it imposes significant costs on existing fossil fuel operations. Oil and coal companies would lose billions of dollars in value because they would both have to pay the tax and experience a surge in new competition from alternatives that can now underprice them. I don't regard this as a problem -- the writing has been on the wall for a long time now and if you've invested in these industries you could have predicted this was coming -- but it is an inconvenience because it causes those industries to lobby against it aggressively, which makes it harder to enact. (But the same will be true of anything with near-term effectiveness, because the whole point is to put them out of business.)
Produce a good in a country without the carbon tax, repackage it in a country that doesn't keep good origin manifests, lie about doing that, then send it to your destination on a solar powered boat (or whatever has the lowest carbon footprint).
We see the same game being played with slavery and child labor. Pick a favorite brand, add "slavery" and you'll find decades of shocked outrage and "commitment to do better"
Mind you, I still support broad carbon taxes. But this is a predictable outcome.
https://www.discoursemagazine.com/economics/2022/01/05/the-c...
https://taxation-customs.ec.europa.eu/carbon-border-adjustme...
For example, imagine a company manufactured something in Qatar (which has the highest per capita CO2 emissions), then they ship their product to Greenland, which has the lowest emissions per capita. Then they repackage their stuff and upon import to the EU, report that "yup, 100% a product of Greenland". Which will almost certainly have low or no carbon tarrifs.
(Not saying any of these countries would actually participate in such a scheme, just an example).
This sort of wheeling and dealing is how companies skirted US and China tarrifs.
My suggestion in another thread is to impose general tariffs on countries that don't have their own carbon tax, which, if it successfully pressures them to adopt one, avoids these kinds of games because now the high-emitting country charges the carbon tax itself regardless of whether its exports make a pitstop in Greenland, which it may do even if some of its exports were avoiding the tariff that way, so that all of them can avoid it.
But it's not obvious this is even necessary because if the largest economies (especially the US) did this, it would shift the economies of scale in favor of non-carbon energy sources to such an extent that they would become cost competitive on their own nearly everywhere else in the world anyway.