update 2: The Linux community has suggested that I use a tar file to backup, as this preserves symlinks. With that, the home directory now takes up just 290 ish GiB, as it should. Now I will be distro hopping, wish me luck!

update: I was able to copy it! There are still some folders that are really big (as many have said, it is probably because symlinks aren't supported in exFAT. When I transfer these files over to btrfs, will the symlinks come back or are they permanently gone?) but, with the uninstallation of Steam and Kdenlive (each taking a ridiculous amount of storage), removing a couple games I don't really play, and removing old folders that lingered around from already uninstalled programs means I now have enough space to fit my home folder in the SSD (like 23 GiB left, so the lack of symlinks still hurts, but still, it fits!)

When running

rsync -Paz /home/sbird "/run/media/sbird/My Passport/sbird"

As told by someone, I run into a ran out of storage error midway. Why is this? My disk usage is about 385 GiB for my home folder, and there is around 780 GiB of space in the external SSD (which already has stuff like photos and documents). Does rsync make doubly copies of it or something? That would be kind of silly. Or is it some other issue?

Note that the SSD is from a reputable brand (Western Digital) so it is unlikely that it is reporting a fake amount of storage.

EDIT: Wait, is it because my laptop SSD is BTRFS and the external SSD is exFAT? Could that be the issue? That would be kind of weird, why would files become so much more bigger with the external SSD?

Thanks everyone for your help to troubleshoot! It was super helpful! Now I need to go to bed, since I've been up so late it's already tomorrow!