By the rules, no
There is no RAW limit to the mechanical complexity of the trigger. One can have a trigger that outlines virtually any set of possible events to happen before your next turn: "if both knights are on the bridge and none of my allies are on the bridge, and the knight closer to me falls prone" is a perfectly valid trigger.
However, the rule states that as a response to the trigger, the player chooses an action - not a set of actions. This means one can't choose to pick of several actions based on any circumstance - either the trigger is fulfilled and the action can be performed as a reaction, or the trigger is not fulfilled and the action can't be performed.
It's not an unreasonable rule change
As long as the complexity of the conditional structure is kept reasonably simple (eg. single action when the trigger is fulfilled, another action if it isn't), I doubt allowing conditional reactions will unbalance the game. Using Ready can result to one's action being wasted if the trigger doesn't get fulfilled, which is understandably a letdown for many players. Allowing a fallback use for the reaction before one's next turn begins can alleviate this problem.
However, if the group has heavy min-maxers, this change can reward being a bit overenthusiastic with Readying to prepare for unlikely circumstances. For example, the party Paladin can declare all their actions as "I Ready to heal Brian the Barbarian with Lay on Hands if he falls unconscious, and attack if he doesn't". This gives the Paladin a lot more tactical versatility than they would have otherwise, as they can instantly revive an unconscious target at only the cost of a lag of one round when making their attacks. As usual, discuss the rule change with your group before trying it out and when trying it out, revise it as necessary.