I found a workaround, although it does not solve the problem completely:
- Open Tools > Customize > Commands
- Select Context menu: Other Context Menus | Server Explorer
- Remove annoying New Query
- Click Add Command...
- Locate Data > New Query Connection...
- Click OK to add it to Server Explorer context menu
This adds New Query Connection... context menu to all Server Explorer connections (as well as the root Data Connections item):

This will fire up the T-SQL Editor.
The only problem with this workaround is that, invoked this way, T-SQL Editor has no way of knowing which connection was selected so it asks for connection settings again. You will also have to specify the database in the toolbar (easy if you're working with just one).
Although it's not perfect, I find it a better solution that constantly hitting the Query Designer awkwardness.