DMCA counter claims do not force a restore, it just permits the host to restore without personal liability for infringement (as the uploader takes on the full legal liability for the material, and often need to identify themselves in the process).

No. The original claim also removes liability for taking down the material. But if a counter notice is filed the material must be restored after no less than 10 and no more than 14 days or liability resumes.

> There’s plenty of exceptions like the DMCA forcing a host to restore access to something if a counterclaim is filed.

The DMCA doesn't do that.

The DMCA allows a host to do that without liability for infringement, and continues to provide the host immunity to liability for any claims the user would otherwise have for the original takedown so long as they do that (the second half is the closest thing to force, but it usually is immaterial because hosts usually structure their contracts with users so that they would have no liability to them to immunize against, which is also why hosts are much more likely to provide reliable notice-takedown than counternotice-restorartion.)

That’s just false:

No liability for taking down generally

[...]unless[...]

C) replaces the removed material and ceases disabling access to it not less than 10, nor more than 14, business days following receipt of the counter notice [...]

USC 512.g.2.c