Is that correct? Is there another benefit?
An offline Root CA sacrifices convenience to gain security.
But, anyway, CA must issue new Intermediate CA certificates and revoke
the old ones... so the only benefit that I can find is that CA issue
different Intermediate certificate for different purposes.
Yes, in case of a compromised Intermediate, the Root CA must be used to revoke old and issue new certs... however, as you note, we're assuming
the Root CA is offline
so, unlike an Intermediate CA, you can't simply connect over the network, submit the CSR, and get the certificate back. "Offline" in this context usually means "air-gapped." Someone needs to pack the CSR for the new Intermediate onto a USB disk, walk to the server room, sit at the keyboard in front of the Root CA, and perform the operation locally. The new certificate needs to be put on the USB disk and carried back out, then connected to a networked system to allow it to be uploaded to the Intermediate CA.
Doing that when Intermediate certs need updating is not difficult. Doing it in any volume becomes impractical, which is why Root CAs generally don't sign individual certificates.
From that point on, the Intermediate CA begins signing certificate requests, but in online mode - network connections transmit the CSRs, and network connections distribute the certs.
So the difference is that the Intermediate CA is online for fast, convenient servicing of requests. The Root CA is offline for slow, awkward, but more secure servicing of requests. The use of multiple Intermediate CAs allows the "risk" of having the authority online and accessible to be divided into different sets of certificates; the eggs are spread into different baskets.