The problem is fundamentally the fact that it is a visual based system.

It is generally faster to communicate orally and process communication aurally than by text message when the message is short enough and requires immediate attention. This is also why urgent alarms (such as those provided by the Gound Proximity Warning System (GPWS) or Traffic Collision Avoidance System[1] (TCAS) have such a component). Some stall prevention systems are even partially tactile based (making a pretty unmistakable shaking feeling (it is loud as well)).

It is incredibly slow to type and then process that visual information. In addition, it's also just much more reliable.

For time critical situations, it's not a viable option.

[1]: Yes, TCAS has a visual component and many alarms do too, but the RAs are auditory and give specific, to the point, instructions on what to do ("climb", "descend").