We already have some systems in place for this, specifically stuff like ACARS and CPDLC. I suspect the issue is in congested terminal airspace, though; if you have many aicraft talking to an approach controller it is not uncommon to have multiple transmissoins at the same time.

However, approach phases are exactly the same periods when the pilots probably shouldn't be looking at their FMCs and texting. Voice communication is faster, and so is acknowledgement of voice commands, while allowing the pilots to concentrate on more critical controls. For example, in many airports it is possible to get departure clearances and such via text, but take-off clearance and taxi instructions will still be voice over radio.