I think this may get very confusing in play. Imagine an X/2 with a tap ability, perhaps a pinger that costs mana to use. I've been using it some turns but not others. Start of my turn, I untap a bunch of guys, and they each take 1 damage. I cast a couple of spells, then declare an attack, which gets complicated with combat tricks. I decide to use my pinger, and the question now is "Has that guy already taken damage this turn?" The answer depends on whether I untapped him way back at the start of this turn... i.e. whether I tapped him last turn. That could often be a long time ago, too long to remember.

With careful bookkeeping it won't be an issue, but not every player is completely careful with their bookkeeping all the time.

I can understand what you're saying, but I feel like the player in control of the Fiend will probably make sure to point out all the damage as it happens. Nonetheless, it may be an inappropriate Rhythm-hoser if it just adds to the "fiddly-ness" of the mechanic.