Solar wind strips that atmosphere very slowly over geological timescales. If we terraform Mars, then we probably have the technology to replenish it faster than that loss.
...because we don't need to? Earth is already full of plant life. Mars isn't.
Sorry for the overly blunt answer, because I know this isn't the question you're asking, but IMHO the question you're asking is based on a ridiculous premise.
We are polluting the Earth. Yeah, this is really bad for the Earth, and for us, but it isn't going to kill all life here. The thing that a "backup planet" is here to protect against is something like being struck by a meteor, or a plague, or nuclear war. Something where everybody dies very quickly, not something where life changes radically.
If you want to terraform Mars, it's probably not because you decided Earth was beyond salvation; I haven't heard many people wanting to abandon Earth. Most people in favor of terraforming Mars want to have a 'backup' for intelligent life, or want to begin humanity's expansion past the bounds of Earth's atmosphere. The 'backup' motive is comparable to suggesting someone backup their electronic data instead of just trying to get very reliable drives.
I am also not sure that it is possible to 'terraform' Earth. 'Terraform' means 'to make like Earth', and Earth is as Earth-like as is possible, no matter how hot or cold it is.
Humans have been terraforming Earth for at least the last hundred years (industrial scale) to 10,000+ years (agriculture, livestock, extinction of other species).
Without a magnetic field, there is way too much radiation on the surface of Mars. A colony would pretty much need to live underground. If you are going to live underground on a planet with unusably-thin air, why not live on the Moon instead?
Instead of just asking the question that many people think they have answered, why not say what you don't like about their answers?
For example, Elon's speech about Mars colonization gives one answer. Maybe you've got great reasons to disagree with him. Maybe there's someone else you'd like to reply to?
(Hint: folks advocating for Mars think the atmosphere is not unusable.)
Not quite sure what you mean: I was responding to a question asking why it would be nice to have a liquid core. In other words, I was commenting on the radiation question.
To hide from Radiation, one plan is to have 16 feet of soil on top of the habitat [1], and limit outdoor time to just 2 hours a day. Given those levels, I'd argue that it would make better sense to build entirely underground on the Moon. The Moon has a shallower gravity well, much stronger solar flux for power, and is much closer for resupply.
Without a spinning core, Earth would be like Mars.
Of course you can live in caves instead of trying to terraform Mars, but doesn't sound so fun.