The DMs issue was that he wasnt prepared to explain to the PCs what was triggering the initiative check. Why ready an action to do something, when there is no information to go on as to why you need to?
My point being that (just by the name of the feat- Alert) the implication is that the character does get some information just before the round begins.
Example (how not to do it from a DM perspective)
DM (noting hidden monsters in ambush): OK, as you walk through the forest, suddenly the whistle of arrows can be heard in the air, and you see movement from the trees all around you. A black fletched arrow thunks into a tree near the Wizard. You're all surprised. Roll initiative!
At which point we should note that the arrow gives an indication of direction - everyone else may be sussing out that point, but the Alert character gets that instantly, and probably should be told where, along its line of flight, the enemy might be hiding.
The enemy is still hidden or obscured - you can't directly target them individually with spells, and you'd be at disadvantage if you attacked before they show themselves.
Or, alternatively - there's noting that the arrow implies an attack has already been made. Does the attacker have the ability to attack and hide in the same round? If not... they are standing right there, where the Alert person can see them...