This is flawed thinking. Surely you're not proposing to implement every support channel your users might want to use? People are generally happy to use designated channels, as long as it's discoverable, easy to use and effective.
People will call you out on Twitter. You likely have a brand account on Twitter already so you'll already get those mentions. Even if you don't have an account people will create a hashtag.
If you want to control your brand perception you don't want the first result when people look for your brand on Twitter to be people tweeting about how terrible you are.
The reasons to do support on Twitter have less to do with what's tech support needs and more with marketing and controlling your brand perception.
