Human CO2 emissions are so far off the scale of anything that has ever happened naturally that nature as she evolved on Earth has no way coping.
There are some questions about how long it stays there, induced seismicity, etc. but the way you're going to find out about this is by doing.
Even if such a ridiculous "Star Trek" miracle device were possible to build with today's technology, it would just ruin the economy.
I've heard of a method that revives native forests rapidly: https://daily.jstor.org/the-miyawaki-method-a-better-way-to-...
First, the typical American carbon footprint is like 15 tons. Which means that a successful backyard bioreactor will have many tons per year of captured carbon to move and dispose of.
Second, the amount of air that needs to be pulled through the system to even have access to 15 tons of carbon is /enormous/. Because CO2 concentrations in air are still very tiny. So a working system pulling a significant portion of one person's carbon footprint will require a really large fan and air management system.