No, carbon capture powered by fossil fuels releases less carbon than it captures. For direct air capture, it takes somewhere around 2,000 kWh to capture a ton of CO2 [1]. You can generate those 2,000 kWh using natural gas and release about 0.4 tons of CO2 (about 0.8 if you use coal). [2] You can be even more efficient by burning natural gas in the sorbent re-generator (avoiding inefficiencies of power generation and transmission).
Out of curiosity, do those figures account for the carbon released during manufacture of the carbon-capture and power-generation equipment you're using? Do they include the carbon emissions of extracting, refining, and delivering the fuels, as well as the emissions from actually using them?
There have been some detalied life-cycle analysis of certain carbon capture methods/plants. As one of the most famous (& funded), Climeworks have had a third party study published to break this down[1]
The machines must result in more capture & storage than they produce - net negative 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.
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.