My 2 cents on Twitter support is that it isn't a support channel, but it should be monitored by the business to make sure users issues are quickly addressed.

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.

Twitter is not chat support and should not be used for most purposes.

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.