tbh i do think it is stronger, axes are like spears if you reduced the damage a little and removed the "must eat" part. it is great against every class in the field and 1 more axe can be exactly what you need to get even stronger class. for example: it takes 2-3 hits with throwing axes to kill bone-breaker, and if you miss one or more of your axes you will most likely die. but this one extra axe gives you a spare shot in case that you missed. another example is fighting an archer, it takes the exact same amount of axes to kill an archer (due to the fact that he has the same armour tire) and archers unlike bone breakers can shoot you from range and they are most likely harder to hit due to most archers moving while shooting. if you get stuck in a fight against an archer and you cant go and melee him (he got something between you and him) missing a shot with an axe simply remove your chance of killing that archer and in the good case you will have to run away to restock your axes (just like you have to do if you miss your forth axe) but having 4 axes gives you one more shot to kill him and if you mised the firs one due to any of the million reasons that are not lack of skills (lag, bad connection, someone else got hit by that axe, your stupid teammate decided that runing infront of you is super important) you will get your fair chance of finishing off the archer. more than that, if you threw an axe for any reason and than an archer approach and demand to get an axe to the face. that axe that you threw eariler will still allow you to beat that archer.
the only negative disadvantage i can see here is fighting against shieldman (you cant melee them now) and fighting spearman in close battle (they can combo you).
so i think that the advantages that the forth axe provides are stronger than
