While I did play Arcane Denial even in some hard control decks (I believe I used it in my Forbidian deck), my most extensive use of the card was in control-combo decks. And here's the past where I get all Comboist Manifesto on this thread and blather on about theory...
While I don't recall seeing this theory spelled out meticulously, others have definitely hinted at it in the past. This relates to cases in combo deckbuilding and gameplay that involve access to cards and the potential for opponents to access cards. Specifically, there is a difference in how card access can be looked at in combo decks relative to other archetypes, and this is of special significance for cards that give both players access to more cards. Since I've never seen anyone else give it a name, I'm going to call it, let's see...
Timetwister Theory! Because of the card Timetwister, of course. Timetwister is one of the Power 9, having a reputation as one of the most powerful cards in the game. It is a mainstay of combo decks in Vintage, where it is still a legal card, albeit restricted. The card's effect is symmetrical: both players shuffle away their hands and graveyards and draw a new hand of 7. So you get a fresh hand of cards, but so does your opponent. Worse still: you pay the mana for the spell, so your opponent is getting a new hand for free. From the perspective of a control or aggro deck, this is a problem. The effect of Timetwister is quite pronounced for only three mana, so one could imagine cases in which it might be good utility, such as an aggro deck using Timetwister to refill its hand after dumping everything, or a control deck grinding out the game to establish a strong board position, then using Timetwister to get back the spells from its graveyard. But paying mana to give the opponent cards is still an issue there, which is part of why the card is essentially nonexistent in full control or aggro decks (the fact that it's a $1000 card probably has some bearing on the situation too, but shut up). And that's where combo is different. Combo decks in Vintage are all too happy to use mana-producing spells, float mana, then fire off Timetwister from a mostly emptied hand, get seven more cards and find something powerful to use the floating mana on, hopefully winning the game before the opponent even gets another turn.
Now, even for combo decks, using Timetwister does give the opponent a new hand. In Vintage, Timetwister can happen so fast that it might take away a good hand that an opponent was hoping to keep, replacing it with random cards, and sort of "unmulliganing" the opponent. While that does happen, the card is nevertheless consuming mana and giving the opponent a fresh hand of 7 cards. Overall, this can still be bad. Both players get new cards and you're the one paying for it. Yeah, either player might get a better or worse hand off Timetwister, so sometimes you're paying mana to give your control opponent a better chance at having answers ready to stop you, or to give your aggro opponent a better chance at having the threats needed to outrace you. But combo decks inherently break this symmetry. Timetwister might give the control player answers or give the aggro player threats, but it can give the combo player cards with which to win the game. If both players are drawing cards off something, your opponent can get utility, but you can get a chance at victory.
The part where the opponent gets to draw cards is still bad! You'd rather not have your opponent get a chance at drawing the card that might stop you. But even if the opponent is getting something, it's not as good as what you're getting. And that's how combo is different. When another player draws a card, it might be a card that brings the player closer to victory by presenting a threat to the opponent's life or by disrupting the opponent in some way. When a combo player draws a card, it might be a piece of the puzzle that wins the game outright. In a sense, your cards are more valuable than your opponent's cards, and drawing more cards, while valuable for either player, is better for the combo player. Timetwister is an extreme example, but the same principle applies on a smaller scale to other cards. This even applies to Arcane Denial.
If one accepts Timetwister Theory, it is still a bit of a stretch to value the measly one-point "slowtrip" from Arcane Denial, especially when it gives the opponent two cards. But the important part about Arcane Denial is still that it is a very cheap hard counter. In a control-combo deck, a deck that is using substantial control elements, but hoping to ultimately set up a combo finish, I contend that Timetwister Theory applies at least a little bit. Yes, letting the opponent draw two cards is undesirable, but being able to stop the biggest, most important threat and then drawing a card for it is good. And since we're combo control, we value drawing a card more than we value not letting the opponent draw a card, and sometimes (although on average, not quite) more than letting the opponent draw two cards. Remember, if we're countering something on the opponent's turn, the cards are being drawn on our turn. As a combo deck, that might be the last turn. We might already have Grim Monolith and Stroke of Genius, with Arcane Denial helping draw us into Power Artifact. Meanwhile, our opponent might draw lands or creatures, making those card draws irrelevant. Yes, the occasion on which Arcane Denial happens to give the opponent the right instant to stop us is unfortunate, but we were already firing off Arcane Denial to stop something, so if our luck is just that poor, with Arcane Denial blocking one must-counter and giving the opponent another one, we were already in a bad spot. After all, the same thing could happen with Timetwister.
And if we're using Arcane Denial on our own turn? In a control-combo deck, we're probably protecting our combo, so the card draws might not even happen anyway, and we're basically using a Counterspell except it costs 1U. In the less likely event that we're casting Arcane Denial on our own turn, but it isn't the kill turn, it's still the case that we protected our combo, and while we do give the opponent two card draws for more chances to disrupt our plans, we also draw one more card, and it might be one that wins the game.