So yes carbon capture using fossil fuels isn't outright impossible because of physics, but practically speaking it's not real because the rest of the chain lowers the already marginal efficiency.
That's not to say that I think DAC is a silver bullet or something - it's obviously not because the CAPEX costs are huge at the moment.
But that doesn't mean we can't also try to accelerate the removal of carbon from our atmosphere.
Should you turn on the machine? No, you should leave it off.
Only when your power sources are green should you turn on the machine.
[1]: https://www.wri.org/insights/direct-air-capture-resource-con...
[2]: https://ourworldindata.org/grapher/carbon-dioxide-emissions-...
For example, using a machine to heat waste biomass with fast pyrolysis turns the biomass into bio-oil. Bio-oil can then be pumped underground resulting in long-term, stable storage. The machine produced far fewer emissions than were captured naturally by the plants.