Not necessarily. A car by itself does not have a big carbon footprint at all, the problem with cars is that basically everyone has one, uses it for everything, and gets a new one every X years.
I drive a small car that I own since 2008, at that point it was already used, and it has 85000 km on it. If everyone of my age and up did that and drove so little, we'd have like 20-100 times less carbon footprint from cars.
It wouldn't even make sense for me to upgrade to an electric car. Yes, it'd use less and greener fuel, but the amount of carbon it'd cost to produce probably wouldn't "break even" until soooo many years in the future, maybe even after I die, at which point it'd even be a bad investment, carbon-wise.
So, like almost anything, it depends.
I would argue that this is not even close to true.
As an open source developer myself, my contribution to open source is pretty much exactly proportional to how well I'm feeling. Getting a donation makes me feel appreciated, using the money makes me feel better, all leading to me being more motivated to spend time on some open-source.
Obviously, being able to only work on open-source would generate more output, but the psychological impact of feeling appreciated to output can't be dismissed and is huge.