A solution without tune2fs.

I'm also aware that the system reserves space for the root user […]. I can write files as root, but I don't know how to make use of this fact.

The default 5% does not mean "5% for root", it means "the last 5% for root, regardless of who uses the rest". So if root manages to use up at least some of these 5% then you need to remove enough files (root's or anyone's files) to actually raise the free space (available to root) above the threshold; only then a non-root user will see space left and be able to use it.

What you deleted apparently was not enough. Delete more.

Note if you delete regular users' files and root takes the reserved space again, and you delete some other regular users' files, then the root will be able to take the resulting free space as reserved space again. It doesn't matter root used the reserved space in the past and has freed nothing since.

I wrote "5%" because it's the default. The answer applies to any non-zero reserved space.

A solution without tune2fs.

I'm also aware that the system reserves space for the root user […]. I can write files as root, but I don't know how to make use of this fact.

The default 5% does not mean "5% for root", it means "the last 5% for root, regardless of who uses the rest". So if root manages to use up at least some of these 5% then you need to remove enough files (root's or anyone's files) to actually raise the free space (available to root) above the threshold; only then a non-root user will see space left and be able to use it.

What you deleted apparently was not enough. Delete more.

Note if you delete regular users' files and root takes the reserved space again, and you delete some other regular users' files, then the root will be able to take the resulting free space as reserved space again. It doesn't matter root used the reserved space in the past and has freed nothing since.

I wrote "5%" because it's the default. The conclusions apply to any non-zero reserved space.