ONLYOFFICE accuses "Euro-Office" maker Nextcloud and IONOS of License Violation
(www.onlyoffice.com)
You're misunderstanding the point of AGPL.
Regular GPL software CAN be run over a network, but because the binary of the software isn't distributed - only an interface is provided to the software itself - the host isn't obligated to provide the source code. A lot of software hosts used this loophole to get around sharing their modifications to GPL licenced software, killing the main point.
That's why AGPL was developed - to protect hosted software. AGPL requires the host to provide source to anyone who has access to the service, not just the binary.
GPL - if I have the binary, I must be granted access to the source
AGPL - if I can access the software, I must be granted access to the source