Everyone has an opinion, I will at only say Grant's record at West Point was spotty at best, possibly because he was bored in a peace time Army. Some military men find their true talent actually fighting wars rather than studying about it.
Then that's part of the question, isn't it - in what way does Kearny's talent show through in 'actually fighting wars' that shows him as a
professional?
By my count he exercises meaningful command in battle at Williamsburg, Seven Pines, Glendale, Malvern, Second Bull Run and Chantilly.
Williamsburg sees Kearny's division turning up as reinforcements that rescue Hooker's division from the problem Hooker created. This is a march of nine miles in seven hours over bad roads (creditable, though not exceptional) and accounts by Ward and Strong indicate that on multiple occasions Kearny ordered fragments of regiments (less than all of Ward's regiment, three times; three companies of the same regiment under Strong, also three times) to charge fortifications, while the 40th NY reports that the left wing of his command was led forward in a charge by Kearny without the colonel being informed.
This isn't Kearny blossoming into a good battlefield leader. Operationally he's okay but tactically he has a preference for the exciting small unit charge over actually organizing the battle line.
Seven Pines sees Kearny overriding Heintzelman's orders (to Birney) to move by rail to the front and reinforce Keyes, which sees Birney court-martialled for disobedience of orders. Kearny then grabs the left wing of the 37th NY and leads it in a charge without telling the regiment's colonel. This forms a hole in the line of Berry's brigade and forces the 37th NY to fall back, complicating Berry's conduct of his battle (he says he could have driven the enemy further back had that not happened).
At Glendale Kearny takes the wrong position, forming north of where he was originally meant to be, and this is part of why McCall's division is in the front line; also during the fighting, Kearny can't be found to approve moving his reserve brigade (Berry) to support McCall, which is part of the reason why McCall's division gets overstrained.
At Second Bull Run, Kearny outright disobeys operational orders (they were poorly formed by Pope, though, overworking the unit) and then (per Return to Bull Run) at the battle itself he drags in a brigade of 9th Corps instead of committing his own (unused) reserves.
Chantilly may (emphasis may) see Kearny disobeying orders to actually go there, but if he did disobey orders then this was a good decision. When he does turn up at Chantilly, though, he doesn't form his division in line of battle. Instead, he turns up at the 21st Massachusetts, a regiment from a completely different
corps (and not even from the 9th Corps division that lost its commander earlier that day) and orders them to advance with wet cartridges on pain of being bombarded with artillery.
This is when Kearny dies.
Kearny is clearly personally brave, and his men march to the sound of the guns on more than one occasion (though other times they
don't march despite orders, like during the lead up to Second Bull Run). He does, however, seem to have considerable trouble doing the difficult work of organizing and controlling his division and regularly goes down to the level of ordering regiment or sub-regimental charges personally, despite being a division commander.
I'm not sure how one can conclude that this man is anything really above average as an AoP division commander in 1862. His one standout quality is aggression, but the fact he keeps ordering small-scale charges that disrupt the battle-line suggests that he has been promoted above the level of his span of command. He might well be a good colonel.