Theoretically carbon capture can work, but just like you said, it takes additional energy to capture carbon, and that amount is more than what it takes to produce the needed electricity if you're using a carbon based energy source.
That said, if you go for something like nuclear, than you do get a clean source of energy that can be used to capture existing carbon. But we're already at the point where our energy infrastructure is inadequate for just electrifying what we currently have, and in a few years the Pickering plant is going to have to shut down due to being so old (though apparently the government is trying to delay it as there's no plans for building a new plant of any sort to replace the Pickering plant).
So even in the best case scenario, it'll be more than a decade before any sort of large scale carbon capture scheme can even be started, as that's how long it'll take to build enough new plants to cover existing demand, let alone accounting for future demand.
What we need isn't thousands of detached single family homes, but hundreds of low and mid-rise buildings that each house dozens. There is no system in the world that'll make single detached homes viable for the entire population. Not to mention that suburbs cost the government more in taxes than they take in, whereas high density neighbourhoods with mixed use buildings are second in economic revenue to downtown cores while providing massive amounts of housing.
I work at a place that spends over a million a year in rent because it uses space from the mixed use first floor of a 30 floor condo. There's dozens of stores like mine that do the same in the area. Imagine how much property tax the city gets from this? How much money must pass through each and every store to be able to afford such rent? And how pretty much every store in the area is doing pretty well despite stores just a few blocks away are crumbling and dying off because there's almost no housing in the area unlike this neighbourhood.
People wanting detached homes is fine. But what about us that don't care about such things? Why don't we get an option for a small but low cost home?