Why is axe a projectile-based, mid-ranged, main-hand weapon to begin with?
The limits of the game engine when it comes to projectiles are well-known after all these years the game has been out. A weapon that fully relies on projectiles is inherently worse. That's why scepter/dagger doesn't see play, and scepter/pistol does. Speaking of scepter, it's the last weapon thief received, and it's mid-ranged, (mostly) projectile-based, and main-hand too. Considering that Thief has had same two off-hand weapons since the release of core game, an off-hand axe would have been a much better investment.
Thoughts on the current design
It's been mostly said by others in this thread, but I'm going to write it anyway:
projectiles can miss any moving target, and increasing projectile speed might still not be enough (target under swiftness, superspeed, or target about-facing)
on some terrain axes will go to Narnia and cannot be pulled back
this is the first Thief weapon that builds damage potential when staying in a place, and drops damage potential when moving away from the place
this is the first Thief weapon you hardly can burst with (in competitive)
there is no impactful soft CC in the kit (compare it to scepter/pistol, how it can immobilize before main damage and apply slow, or chill with off-hand dagger)
there is no hard CC in the kit
there is no boonrip in the kit
the only defensive resource is 0.5s evade on #3 (which is there mostly to prevent getting interrupted during 0.5s cast), and the evade is stationary
there are no other resources that help with survival (no cleanse, block, reflect, shadowstep, leap, blast, or movement skill)
lack of aforementioned defensive resources doesn't fit the playstyle where you are meant to maintain range and kite
#3 is not piercing, easy for it to be body-blocked. If cleave is the idea, this doesn't match it. EDIT: my bad, it does pierce, just missing in tooltip
#3 is clearly telegraphed, easy to evade or block it (more about this below)
hardly a viable option for Specter
Pointless split between damage types
Condition damage with dagger, power damage with pistol. So if I don't enjoy playing one damage type, and I want maximum potential out of a damage type, I'm locked into playing with a specific off-hand weapon? I find this extremely irritating.
Lack of synergy (especially with off-hand pistol)
With axe/pistol you can't blast, you can't leap, you can't reliably blind. Skill #3 with either off-hand weapon has such a clear tell that every enemy can avoid it as long as their monitor is plugged in. There is nothing on the kit to bait the resources of the enemy. There is nothing on the kit to increase the chances of landing #3, even if the projectiles were faster. Initiative costs are high as well - #3 costs 5 initiative in PvP, #5 costs 6. You're out of initiative almost instantly, all while you're very likely to miss #3, and that's assuming you had enough initiative before. Moreover, using #3 consumes your axe stacks, so it's easy to tell what thief is going to do next in close range: either follow up with #2, or try to stealth with #5.
Suggestions/Ideas for rework
Expanding on the idea of redirectable spinning axes (if you must), to provide defense, axes could spin around Thief, but they could be used offensively, so by throwing them towards target or pulling them back a Thief would exchange defense for damage or vice versa. Optionally, while the axes are spinning around the Thief, they could do cleave damage in melee range. Another idea for defense could be picking the spinning axes up from ground, which would give you an evade frame, just like Mirage's Mirrors. For damage, perhaps you could move spinning axes like Ventari's Tablet.
A simple solution for increasing momentum in current implementation could be to lower the maximum amount of stackable axes, but increase the damage. Alternatively make the autoattack throw multiple axes instead.
As a closing note
For me Thief encompasses movement, speed, timing, high risk and high reward, manipulation of momentum in competitive setting. I don't see it as an annoying mosquito, bullying enemies, I wish it to be a force to be reckoned with. This weapon doesn't feel like Thief to me. It doesn't fit the playstyle and identity. Personally, I dreamed of playing dual axe in melee range, blending in a playstyle of a bruiser. Ranger's dual maces come to mind. I don't expect my vision to match anyone else's, of course.
Anyway, I only hope that whoever has worked on this weapon for Thief, or will be working on any future ones, has "fun to play" and "fun to play against" on top of their priority list.