Patron Warrior for bottom tier.
It sounds like you aren't mulliganing properly. Arcane Shot, Bear Trap, Freezing Trap, Snipe, King's Elekk, Animal Companion and Lock N' Load (although it's way better earlygame if you have the coin). That's 11 earlygame cards, compared to Mage's 13.
But they don't compare favorably to the mage early game, is the problem.
Like, the common start of turn 1 lowly squire or the torunament medic. What do you do? Arcane shot? Then fallen hero/boneguard liuetenant followup get stronger. If you let it sit then it will grow in power, the elek gets killed by these early on. Animal Companion is RNG and as such doesn't always suit the board. Leokk rarely does much here, Huffer trades horribly with the early game and hitting the face is not the aim for this deck especially with the card that heals both players, so Misha is typically the one to want. The traps being static make them much easier to play around, snipe is usually the strongest threat but the 2/1 divine shield is a hard counter to it.
In the mid-late game it still is fairly ugly. Warhorse joust has a HUGE impact on the match, the divine shield taunt is basically polyboar bait and the bigger minnions get eaten by flame lance. Acidmaw is unbelievably bad in this match since the mage hero power becomes an assassinate(that can also buff their board!), even if you get uth board clears. Which is another thing, UTH is pretty easy to play around as everyone knows the hunter has hunter's mark alongside it, so the mage just needs to not super overextend.
It's certainly not hopeless but the hunter always plays from behind and if they get into a board control position it's much better, but that's hard to pull off. Lock N Load can help a lot, but even the really nice early plays with coin + arcane shot can still do something like give you feign death and gladiator's longbow...which is exactly what happened to me before.
Oh, and Rhonin is nice vs. this deck since there's zero silence and a 7/7 can't be ignored, making the triple missiles another threat. Double Frost Giant makes the joust sketchy and gives the mage some powerful late game on top of that.
if they had some mad scientists the deck would be way better to hedge through the early game. For now the best start the hunter can hope for tends to be the mage having no turn 1, and the garrison turn 2 for the elekk to kill since that's not a big deal early on.