For the symmetric cryptography (so obviously AES and ChaCha, but also in effect the SHA-2 family) we can hand wave the quantum attacks as halving key length by enabling a sort of meet-in-the-middle attack (this attack is why it was 3DES not 2DES when they strengthened DES). There's a lot of hand waving involved. Your real Quantum Computer won't in fact be equivalent cost to the non-quantum computer, or as fast, or as small, the speed-ups aren't quite halving, and so on. But it's enough to say OK, what if AES-128 was as weak as a hypothetical AES-64, and that's fine because we have AES-256 anyway.

However, the main focus is on Key Exchange. Why? Well, Key Exchange is the clever bit where we don't say our secrets out loud. Using a KEX two parties Alice and Bob agree a secret but neither of them utters it. Break that and you can learn the secret, which was used to encrypt everything else - for any conversation, including conversations which you recorded any time in the past, such as today.

If future bad guys did have a Quantum Computer the Key Exchange lets them read existing conversations they've tapped but today can't read, whereas breaking say the signing algorithm wouldn't let them somehow go back in time and sign things now because that's not how time works. So that's why the focus on KEX. Once such a thing exists or clearly is soon to deliver it's important to solve a lot of other problems such as signing, but for KEX that's already too late.