DMCA is not an agreement, it is a law explaining what content is legal and what content is not. The law doesn't mention banning or deletion of content, it doesn't talk about consequences to breaking the law.Yes, they do. Just as Youtube must delete the channels of DMCA violators. Otherwise they risk becoming liable. DMCA is essentially an agreement made based on Rights Holders understanding that the nature of sites like Youtube and Twitch means that those Service Providers can not pre-emptively prevent copyright violations so rather than immediately going to court over them, the Rights Holders allow the Service Providers some room to remove the violating content. If this *only* applied to VODs and not the users then it's effectively useless to Rights Holders as users would simply re-upload after removal. Or in the case of Twitch, the VODs are miniscule as the service is focused on live broadcast. So that means the users themselves need to removed. Too many DMCA requests against a single user can and would be used in court as evidence that the Service Provider is not doing an adequate job of removing violating material and could result in the Service Provider becoming liable.
All that is required by law is removing the violating content, not banning. Twitch made the choice to suddenly ban everyone with no warning, their choice to ban people has zero to do with the DMCA.
Your example of multiple requests on a single user isn't part of this conversation, because Twitch banned *EVERYONE*, not people who had been warned and ignored the warning.