Actually seems like that is the case from watching the dev stream again, but I could be misinterpreting it.
The way they phrase Adrenaline in the stream is that Adrenaline checks the status of your hand in the moment of playing the card, which they mention when talking about the new Viper Witcher. The confusing part for me is what you are asking in regards to more passive abilities and if they trigger multiple times provided your hand count is within that range, which they don't really mention at all if Kolgrim will trigger multiple times, or just the once.
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If Kolgrim triggered multiple times, that would just play right into tall removal anyway. Unless you play a mill deck and milled or baited all tall removal from your opponent, or just have that huge of a card advantage by some other means, he's near guaranteed to be smitten from the board at that point.
I think Kolgrim does trigger multiple times a turn, otherwise they would have given him a Deploy ability instead of a "at the end of the turn" ability.
He's a big engine at the end of the round, basically. I don't know how viable that'll be, though because as you say he'll get taken care of.