Why was the twentieth century Austro-Hungarian army so ineffectual and poorly equipped if many of the leading scientists and thinkers of the time were Austrian?

In terms of efficacy, I think u/k_hopz has already provided several comprehensive responses I can't add to (via the link provided in prior post).

I will add that a culture that boasts leading scientists and thinkers is complementary to, but not necessarily causal to a strong military (you can have lots of smart people without having a strong military, but if you want a strong military it'd be nice to have lots of smart people, too, is what I'm saying). Indeed, many elements of Austro-Hungarian society were in a phase were it was not fashionable to be associated with the military.

As for the equipment it comes down to money and politics. The Austrians spent less on their military (relative to their population size) than any other European "power" including Russia, who had many more people. Britain spent nearly 4x as much and France and Germany each spent more than 2x per capita (based on 1906 budgets and currency). The cause of this was partly the government's need to spend money elsewhere due to a large and (arguably) overstaffed civil administration (The peace time army had 415,000 officers and men v 550,000 bureaucrats at the imperial level, per Holger Herwig in The First World War: Germany and Austria-Hungary) and an economy that was growing well but still lagged behind other European nations (thus providing a lower tax base).

There was also significant political strife around military expenses due to the structure of the Austro-Hungarian government. There were elements of the Hungarian government that saw the Imperial and Royal (k.u.k) army as a potential tool of future Germanic oppression of the Magyars (looking back to 1848), and so wished to deny it funding or wished to use funding the k.u.k as a bargaining chip for increased money for the Hungarian "national guard" the Honved. Naturally, there were elements of the Austrian government (and, to be fair, elements of the Hungarian government too) that saw the Honved as the core of a future Hungarian Army and tool to break away from Imperial control, and so there was reluctance to increase funding for the Honved. Negotiations on budgets tended to be long, drawn out affairs which could cause major projects to be delayed (such as major overhauls of the artillery). So... Austria-Hungary underspent and took forever to get anything done due to political shenanigans. That's the TLDR.

The low amount of money dedicated to the military had a few notable impacts, that would all be crucial later:

The military didn't pay well, and this had led to declines in those enrolling in the officer corp (attendance at Austrian military academies had fallen by nearly 50% from 1897 to 1913)
The Austrians conscripted a very small portion of the available annual class for military training (0.29% v 0.75% in France, 0.47% in Germany, and 0.35% in Russia). Not only did this mean Austria had a small standing army (relative to population size), this also meant that the reserve pool of men with prior service training was much smaller than other nations
The Austrians didn't do as much training as other powers. Large field exercises were not as common (they were expensive), live fire practice with small arms was restricted relative to other armies. John Dredger, in Tactics and Procurement in the Habsburg Military, 1866-1918 states that, in the artillery branch, the k.u.k batteries fired only 208 shots per year in training (v 464 in Germany, 390 in France, and 480 in Russia).
Austrian artillery was woefully deficient. While GWC can have a youtube video where they claim Austrian artillery was actually just fine, they are totally incorrect. The artillery support in the k.u.k was inadequate in every possible measure, largely because artillery was expensive to buy, expensive to practice with, expensive to maintain, and also artillery technology had advanced quickly in the recent couple decades, meaning it was even more expensive to keep current. Outside of certain specialist guns (like the famous Skoda 30.5 mk11), k.u.k guns were made from a steel-bronze metal notable for being cheaper than modern steel and having a good shelf life, requiring less expensive repairs and replacements overall than steel guns. On the other hand, the steel bronze guns of the k.u.k enjoyed a range deficiency relative to the equivalent guns of their enemies of 1,000-3,000 meters. Austrian guns also did not incorporate modern pneumatic recoil systems to allow for rapid fire, instead relying on an obsolete spade brake. This meant that the typical Austrian field battery could fire at about half the rate of the equivalent Russian (76.2mm 1902) or German (7.7cm FK 96 n.a) made battery and roughly 1/3 the rate of the French 75's (assuming the guns had enough shells, that is). FURTHER, beyond the facts that Austrian guns were outranged and outshot, the k.u.k simply did not have enough guns in the first place. They had no heavy guns at the divisional level (and few enough at the Corp level). An Austrian division would have 42 field guns per division, compared to 72 field guns in a German division. Dredger cites that Austria had 4 guns per 1,000 men compared to Germany's 6.5, Britain's 6.3, France's 5, Italy's 4, and Russia's 3.75. Dredger goes on to state that the k.u.k. also maintained smaller shell stocks per gun than the other powers (Herwig places it at 250/gun v 700/gun for Germany and France).
Austria was also very under-supplied on machine guns and airplanes, for the same reasons as before (money, speed of funding decisions), though this was a trait some other powers shared and not particularly unique to the k.u.k.