If they'd instead looked at - say - the difficulties their men had estimating range, and realized that the reason the French assault columns got close was that the Austrians were firing over the heads of the columns (not adjusting range fast enough) then they could have instead kept a firepower-based method and also improved their aim.
I'm not sure you can expect much more from the Austrians on marksmanship:
we should always bear in mind that the constraints the Austrian army were working under were far more severe than the Prussian. Training recruits to shoot accurately is much more difficult when they don't even know the 80 German words of command to manoeuvre on the battlefield. People also treat training in marksmanship as if it's entirely costless: in fact, the cost of increasing the number of practice rounds available to Austrian troops to 100 per year was the equivalent of the annual cost of the war ministry's Vienna secretariat, or the officer's pensions bill.
The Austrians aren't that far behind the European curve- you mention the
Ecole du Tir in the same breath as Hythe, but actually the French system was nowhere near the British:
it was laid down in a programme issued by the minister of war for the construction of arms, that it was not necessary that the fire of infantry soldiers should exceed 600 metres. The chasseurs alone have movable sights capable of being adjusted for long distances. The Ecole du Tir, the system of musketry instruction, originated also with them; but they seem to have halted at an early stage, and to have refrained from carrying it to its fullest development. The French soldier of the line is only trained to fire up to 400 yards, and instead of using a sight, is taught to make allowance for the different distances by aiming at different points of the target, or different parts in a man's body; and it is evident that they do not consider this method to be of much avail beyond 300 yards. (
source)
The Austrians might have more success if they split the responsibilities post-1859. Train a limited number of men per platoon solidly in range estimation, and have them inform the rest of the platoon what the range is. Simplify the Lorenz
sight, to make it easier for the soldiers to aim. Perhaps even drop the "battalion divisions" system in favour of something less manoeuvrable but less complex. Really, anything that's not adopting assault tactics would help.
Step two - better artillery. The Austrians actually have this already, with RBL guns while Krupp is still working out the issues with his own pieces.
The Austrians have RML pieces, similar to ones the French used against them in Italy. It's probably unrealistic to ask much more of them given how well they perform during the war- as things stand, they're one of the few components of the Austrian army which comes out with an enhanced reputation. Existing flaws in Prussian artillery doctrine should be enough.
Step three - which is a luxury - breech loaders. These aren't really required,
I'm not sure it is a luxury, in the context of Prussian tactics. If the Austrians deploy skirmishers with minie rifles against swarms of Prussian skirmishers with the Dreyse, they're going to get overwhelmed. It might be feasible, though expensive, to arm the Jaeger battalions with a good, solid breech-loading conversion of the Lorenz (or even the Dreyse itself) and the remaining battalions with minie rifles. The problem is that the Prussians can still deploy more skirmishers, and they'll be better-trained than the Austrians.
Of course, it's entirely plausible that the Austrians do all the right things, and the Prussians still adapt to it: for instance, putting more emphasis on flanking to avoid head-on assaults. They do something similar when battle-plans start to fall apart during the Franco-Prussian war.