Sorry 'bout this, but I do re-enactment, and it's Norman era. I can think of one way an axe might be a symbol of authority: Men-at-arms. Your average man-at-arms will be a conscript, doing army service during times of war. He will probably have a spear, an aketon/gambeson and a helmet. That's it. He might have a knife. A spear is even more a common weapon than an axe, the reason being, a spear can be as little as a sharpened stick, possibly with a knife lashed to the end to supply a reliable point. An axe needs a metal head, and metal is EXPENSIVE! If you can afford enough metal to make an axe-head, let alone two (one for wood, one for people), you are doing well for yourself. Metal being expensive is the reason that a specialised warrior class exists. Knights were the tanks of the norman era, and they knew it. In fact, crossbows were banned by the POPE for taking down too many knights. (crossbows were easy to learn. I quote: "a longbow takes years of training to learn how to use effectively, whereas you can teach any idiot how to use a crossbow in five minutes" (this is usually followed by "and this is our idiot")) Longbows, a skilled weapon, never had that problem. (One thing you need to remember is that plate armour hadn't been invented at the dates we re-enact. Mail (not chainmail, that's a Victorian word) was the most heavy armour of the day, and it was too easily broken by arrows - Making wearing it useful, but not incredibly useful.)
The problem with the axe is that it is ineffective on horseback (too top heavy, you drop it), so if horseback warfare takes over you lose it.
Norman knights rode on horseback, and used maces. If you want to say that a lever effect on a large lump of metal on a stick is worse with an axe than with a mace, then fine, but you aren't getting any history marks from me. Incidentally, the mace is a very effective weapon on horseback. Swing it underarm, charging towards your opponent, and you might be able to break their spine, even hitting their chest. Indeed, we only use maces to hit shields, not each other (or at least, not each other on purpose...).
I'm not arguing that axeman + longbow is better than heavy cavalry,
Okay, well, how about spearman + Longbow? The horse will not be too happy about charging the sharp pointy stick, and the longbow wreaks havoc with unified, organized charges. The longbow is the medieval machine gun. Well trained archers could loose (not fire, that's gunpowder weapons) 30 arrows per minute. English civil war (roundheads, cavaliers) tactics involved horses, guns and pikes. A standard battle went one side fires, the other side charges on horseback, the first side closes and lowers pikes, horses retreat, the the other side gets a go.