The problem with this is that you don't necessarily know if you want to react to a specific enemy or not until you see what the enemies do. I may intend to save my reaction to shield against that barbarian that is low on initiative order, but if I see the spellcaster casting some huge damage spell instead of the cantrip I expected he would, I might change my mind and try to shield or counterspell the caster instead, and my choice of which to do might depend on whether the spell the caster is casting is save-based or attack-based. So, combat flow has to be able to stop at any given moment to allow players to make an informed decision on the fly.
This is not how counterspell works in D&D 5E.
All your character would know was that a spellcaster within 60 feet of them was casting a spell. You do not know what the spell does. You do not know if it's a damage spell. You do not know if it is a cantrip. You do not know the level of the spell. You do not know if it is save or attack. The choice must be made immediately with zero additional information.
In your scenario the choice you would have is to use your reaction to counter an unknown spell or save your reaction to cast shield against that barbarian.
Exactly the same information you would have had you clicked a checkbox before combat.
