Cause, as written, the Handy Haversack is:
"When the wearer reaches into it for a specific item, that item is always on top."
The specific item you're reaching for would be the Deck, unless you managed to get an individual card from the deck by itself. The card you are to draw is unknown until you draw it, and you always draw from the top. If you don't draw a card within an hour, the top card is automatically flipped for you, since the words "Deck" and "Dick" are pretty close to one another.
So as written, this doesn't work. Sorry.
EDIT: Though, be careful when drawing out the deck that you don't accidentally just take the top card. After all, the Deck is right on top of the haversack. It'd be a shame if you drew the wrong card from, essentially, one of the primal instruments of chaos.
I bold'ed where you're wrong. The first one is a semantic point, and would be pointless to argue about back & forth. But at no point does the Deck require you to draw from the top. Think the stage magician with the deck splayed out saying "pick a card". That's the deck, you can draw from anywhere (kinda a "nothing says I can't argument, but I've never seen it argued any other way until you just did). As such, you put it sideways into the Haversack. Tell the bag what card you want to be "on top", and it raises the card from the side, thus you just draw the card that is raised. It's like having a trimmed card (to influence the cut), but in reverse.
Because of this, it is not about "doesn't work cause artifacts stronger!", since their effects don't actually directly interact, you're just using one item to cheat another. Think of it this way, if a person had 600 of the amulets that let you re-live a round, and a contingent spell, they would be able to achieve the same thing: draw a card, not what I want, re-do with amulet, if Void, contingent triggers magic mouth to activate the amulet before the void has its effect. This also works. Haversack is just cheaper.
The point is, CO and TO boards are 95% about how to make the Random Number Generator (usually dice) favor you. This is why Iterative Probability is so heatedly debated here, which, if you remember, is the main thing Osle was worried about that list of acronyms.
If you hand a CO board a use of Random Number Variables (in this case, the RNG is cards), wherein there is that good of a side effect, we will (and have) find at least 7 ways to break it. (there may be more than 7 with the deck, I've now covered 2 and don't plan to do more)