Yeah, no. Subreddit mods can't do that, only the sitewide admins.
It's easy to prove this as anyone can create a new subreddit and see exactly what functions mods get. They don't get the ability to issue sitewide bans, only bans from their own subreddit.
Mods don't have any more ability to do it than any other user. They literally just have to use the "report user" function, same as anyone else on reddit. There's no secret special back door access to the admins that mods get. This is literally one of the things that mods complain about - they don't have any way to deal with ban avoidance and alt accounts except to hit "report" and cross their fingers.
Mods don't have any extra way to report users to admins that all users don't have access to. It's literally just the "report user" button.
Mods certainly don't have any way to detect alts beyond just knowing their community and manually spotting posts with a similar style to known troll accounts.
Seriously dude, you can see exactly what tools mods have access to just by creating your own sub and just fucking looking for yourself. The mod tools are shit, they get virtually nothing, certainly nothing like you seem to imagine they have. The fact that the mod tools are shit and have been shit since forever and that the admins have made promises to sort them out but never done anything about it are some of the major contributing factors in the protest and the move here. The API change were really just the last straw.
You're obviously butthurt over getting banned but based on your attitude here you probably deserved it.