Inbox Zero is not about having no emails in your inbox, it's about not thinking about email when you don't want to, because when you DO do email, you catch up entirely.
I fundamentally disagree with the concept of Inbox Infinity, not because it is the opposite of Inbox Zero, but because this person has decided that it is okay to declare that you use a particular communication channel, and then (often) not use it. If you don't want to feel tethered to a particular type of communication, don't use it. Know how to not feel overwhelmed by Slack? Don't use Slack.
You know who Inbox Infinity works for? Anyone who a/ is super popular and gets so many emails that the people emailing them won't expect a reply, even if they don't say it; and b/ Anyone who wants to shirk whatever they want, and not what they don't. Which I realize the article author does acknowledge about 2/3 of the way down the page. And he ends it right - having a system is the most important thing.
I get where Inbox Infinity person is coming from, because I totally agree with the concept that you have to know how to say yes and say no to stuff in order to get what is really a priority done. But if you think email is your saving grace, then you just aren't managing your inbox correctly. Email itself is an agnostic channel, and can be just as good or bad as any other channel. People just let it get out of hand and don't go in and catch it up and fix it for themselves so that they have a clean experience ongoing.
To misquote St. Francis DeSales, use the communications channel you say you use, and use them well.