Just to be sure I understand your point. Are you saying they CO2 in as an alternative to having the greenhouse ventilate to the atmosphere?

I guess the reason being that the plants would convert all the CO2 to O2 in the greenhouse and then suffocate without the vent or pumped CO2.

Is that the most economical way to keep the greenhouse warm enough?

There are heat exchangers which are passive devices you can attached to an intake and exhaust to reclaim heat.

Yes, exactly: https://www.dutchgreenhouses.com/en/technology/co2-enrichmen...

Burning gas in the enclosed space not only enriches the CO2, but also warms up the greenhouse even more.

You made it sound like adding CO2 was only a byproduct of an attempt to keep the greenhouse warm, not part of the intention. Yet your source says

" The supply of extra carbon dioxide is an often applied method to increase the yield of greenhouse crops."

So I'm not sure what your point is.

The point is that adding CO2 is necessary to solve a problem that greenhouses create, the enclosed plants deplete the available CO2 and stop growing. It's not applicable to the world at large.

In an open atmosphere plants aren't limited by available CO2, they are limited by sunlight, temperature, soil nutrients, or usually water availability. Adding more CO2 has little to no benefit for the plants.

"Has plant growth increased alongside rising levels of carbon dioxide in the atmosphere?

It turns out the answer is Yes – in a big way. A new study published in the April 6 edition of the journal Nature concludes that as emissions of carbon dioxide from burning fossil fuels have increased since the start of the 20th century, plants around the world are utilizing 30 percent more carbon dioxide (CO2), spurring plant growth." - https://www.noaa.gov/news/study-global-plant-growth-surging-...

I mean, a little, but the closed nature of the greenhouse is still the bigger problem.

1000 cubic meters of air only has something like 150 grams of carbon in it; even if your plants in question are 95% water, you're still only talking a few kilograms of plant growth before you completely exhaust the air in an enclosed space.