Once the samurai usurped the governing power of the kuge (Shugo replacing Kokushi), didn't tax revenue get redirected to them instead? I suppose some kuge lost their income, but as a class they continued to survive through many upheaval and frankly atrocious time.

That's not how it worked. Shugo were invested with authority over taxation only under the Muromachi government, in the 14th century (specifically, 1352, I believe). Their official authority was severely limited during the Kamakura period, restricted to pursuing and punishing major crimes (murder, arson, rebellion, banditry, piracy, nightly assaults) and to organizing shogunate vassal service in the capital (and, later, in Kamakura as well as on the anti-Mongol-defenses in Kyushu). Hence, provincial governors still were nominally responsible for collecting taxes from "public" lands, i.e., everything that wasn't designated part of a shōen (estate). Of course, by then, usually the nominal provincial governor would just dispatch an agent of his, instead of going to the province himself. Also, provinces during Kamakura times were mostly in the hands of powerful nobles, the retired emperor, or the shogunate, who “recommended” the governor for the respective province (this was a de facto appointment though, since recommendations were followed).

In estates, the local estate officials (geshi or jitō) were responsible for forwarding taxes to the proprietors, who, ultimately, were either the imperial family or some high-ranked noble (e.g., one of the Fujiwara main branches etc.), if they weren’t major Buddhist monasteries or major Shinto shrines (such as Ise Grand Shrine). [edit from here:] The arrangements between shogunate vassals (jitō) and the proprietors were manifold and based on local customs. However, the conflict of interest between these local administrators and central beneficaries often led to conflicts over the arrangements, on which we have hundreds of lawsuits more-or-less well documented available. By the late Kamakura period, the arrangements would, in places, lead to a factual split of the respective territory in question between the local warrior-official and his overlord, who then would send his own agent to manage the rest of it. The estate-system, which formed the main basis of income for the court nobility, survived in some form or other until the Sengoku period (its collapse was, again, sooner and more noticeable in some places than in others: some estates, especially located in the region close to the capital, remained in place for a very long time). [edit end]

The disintegration process of direct court authority over provincial administration was not in the form of some sudden, and total, displacement of the court by shogunate agents, but rather a very gradual affair, with less and less land being under public control, and shugo gaining more and more rights which put them in direct functional rivalry with the provincial governor, eventually rendering the latter meaningless.

That being said, post-Kamakura land administration is not my field, so I’m not certain about the particulars of what exactly happened to the finances of the various courtiers.

Continuing from where /u/Morricane left off, due to the need for military mobilization and the loyalty of the Shugo during the Nanbokuchō, the Shugo were given the right to half of the tax revenue of the province, called hanzei, and the right to settle court cases involving land claim disputes in within their provinces (which had overwhelmed the Kamakura bakufu). The Shugo now were given half the taxes from the estate tax collectors. The bakufu did try to stop the hanzei in provinces where war ended, but such orders seems to not have been followed on the round. In 1368 the bakufu ordered all estate belonging to the imperial family, regent families, and temples to return to normal (cancel the hanzei), but basically made the hanzei permanent for all other estates. In this the bakufu seem to have made formal the situation already on site, in exchange for the return to normal for the estates under those most powerful non-warriors.

With the power of military, judicial, and taxation, and with the nominal estate holders being away in Kyotō (or to a lesser extent Kamakura), the Shugo began working to get the rest of the taxes from their provinces. The method was to get themselves appointed the collectors, of which they would forward a certain amount (depending on arrangement but usually a bit more than half) to the estate holders. Basically for a (large) fee the estate owners no longer had to appoint and pay their own tax collectors. However with Shugo having basically all the power in the province and having an important position in the Bakufu, often they would make an excuse to not forward the taxes. For instance, the temples of Mount Kōya in 1439 brought forth the lawsuit against the Yamana clan that they were supposed to be forwarded 1000 koku from Ōta estate in modern Hiroshima each year since 1402, but by the time of the lawsuit the Yamana were 26,000 koku behind in payment. If that's true, then the Yamana forwarded less than a third of what they were supposed to. Besides lawsuits, the response of the estate owners were to try to get their estates declared off limits from the Shugo.

This power-struggle between off-site owners and on-site collectors continued until the Sengoku. In the Sengoku even the Shugo had problems governing their provinces. With no strong central power to mandate the forwarding of taxes, those on-site basically kept everything, and almost none of the revenue were forwarded to their offsite owners, who weren't much of a owner by this point. Some of the stronger lords went so far as to formally declare themselves the authority before the bakufu with no estates off limit. Even the imperial family suffered greatly. They had over 200 estates that gave them over 7,500 kan in the worst years before the Ōnin. After the Ōnin, income fell to about 60 estates submitting 1,000 kan. Between 1521 and 1569 income was reduced to 690 kan a year (less than 10% of pre-Ōnin). Many times the off-site owners went to their estates to try to get the taxes themselves, but for the most part are not successful. Until Nobunaga and Hideyoshi, the off-site owners basically had to do their best to try to survive on a hugely reduced income, or go live with one of the lords on their patronage, or for the imperial family who must remain in the palace to seek donations and sell titles.

So as you can see the process of the kuge not getting tax revenue is not instant with the Kamakura bakufu. Rather, the process of the kuge getting almost everything to them getting almost nothing took over three centuries.