Email everyone expects is effectively a queue, whereas a Chat request users expect to be answered quickly. If the application ration of users to support staff is too high, the wait would be long for a chat request thereby creating frustration for users which would then consider the support negatively, likely driving up cancellations or complaints.
The terms usually state emails will be answered within 4 hrs for emergency, 24 for standard support etc. This allows the team to prioritize and manage the workflow, versus Chat where it is very difficult to do that.
Phone support likewise is generally set for those high tiers where the ratio of users to staff is manageable so this is acceptable as well. Additionally, phone support might be the comfort of taking the support case, documenting it and sending it to development/support. Which is a different expectation than online Chat has for most people.
Generally I have seen two things happen on twitter, first is one or more users say there is an issue happening, second if the company is quiet then the users become restless and it gets a little punchy.
I have advised clients to make sure they monitor Twitter but don't try to solve customer tickets or issues there, quickly respond so users know you are alive and then move them to a real support channel.
Twitter for businesses should be used for outreach and outbound communication, with direct responses to users when proper.
If I owned a company, I would not 1) want support tickets running public, 2) want support tickets being available to twitter internal with private customer data potentially in DMs.
Twitter is great for outreach though.
