As with all research, the a multiple ways this can be interpreted. What I see from your game is that our mages aren't as different as we'd like. And honestly the solution to the problem is go make them more unique. They need things to differentiate themselves that you can't always just pick for your spellbook. So in the midst of working on PVS, this is a good realization.
I look at the druid and the necromancer and it seems they're a lot more unique. It would be quite tough for the wizard to try and "be them". So then looking at them, I'll be looking at what we can do to give our mages more unique feel.
Now this is not a response directly about balance, but I feel this issue better addresses what has come up from the game proposed here.
I'm really glad that you think this way! It's exactly why I'm pushing so hard for better unique cards such as the good version of conquer. I do have to add though, here's a list of the things I can do better as a Wizard than other mages.
1. Curse Wizard, better than curse Warlock/Necromancer
2. Attack spell fire Wizard, better than Adramelech Warlock
3. Aggro, better than Forcemaster and Joktari Beastmaster most notably, but better than all other mages too.
(Attack spell Wizard, with or without Jinx)
4. 1-big strategy, better than all other mages, biggest names being Warlock and Priest. (See Wizzly Grizard)
5. Turtling for time wins, better than all mages aside from Priestess. (My gencon book last year had me at 40+ life every game.)
6. Mid-game in general, better than all other mages. (Weighted Training Clothes)
And here's a list of things mages can do better than Wizard, and the mages that do so.
1. Auto attacking. Forcemaster and Warlock. Keep in mind I am of the opinion of auto attacking is not currently a viable competitive strategy
2. Massing Creatures. Priestess, Necromancer, and Beastmaster. This also generally doesn't happen because it relies on your opponent not interacting with you.
3. Turtling for time wins. Priestess, also relies on non-interaction.
Here's a list of things I feel are equal.
1. Destroying conjurations, Warlord and Wizard are fairly equal thanks to Akiro's Hammer.
I will add that one game on a stream definitely isn't conclusive evidence, but though my personal testing I draw the same conclusion that Sailor Vulcan does. Curse Warlock isn't even close, curse Necromancer is closer in power though.