Brusilov could be assigned command of the 6th Army, which was kept around Petograd and got into the fight late, or command in the Far East. In empires there are lots of places to post senior officers where no one will have heard of them.
The likeliest alternative to Brusilov at the Southwest Front, Dmitry Shcherbachev, seems pretty competent based on some quick internet research so there may have been no difference after all.
The intention at STAVKA was for the attack by Southwest Front to be a diversionary attack, to draw German forces to the Austrian Front, and then open up the main offensive against the Germans. The southwestern attack turned into the main and only axis of advance because of the initial success. The likeliest alternative is that the Southwest Front still attacks, it is the main diversionary attack as planned, and then the main offensive happens in the north.
So in terms of the military situation or the Tsarist army, I don't think anything changes at all, just the narrative. The Russian advance looked impressive on the map but they didn't take any place of importance and the Central Powers were able to stabilize the front. The Russians took heavy losses, but this would have been the case if there had been two offensives, instead of successive offensives in the southwest.
However, the offensive had a weird effect on the Central Powers command politics. The success of the Brusilov offensive induced Rumania to declare war on Germany and Austria-Hungary. For some reason the chief of OhL, Falkenhayn, was blamed for this and fired. In fact he wound up tasked with fighting the Rumanians, which he did successfully, so its not clear if Rumania entering the war against them really hurt the Central Powers (they got Rumanian's resources anyway and a good deal of Russian forces, including the aforementioned 6th Army, were sent to the new front). Falkenhayn had unsuccessfully tried to stop the diversion of Austro-Hungarian forces from Galicia to Italy, which had been the main cause of the disaster. The replacement of Falkenhayn by Hindenburg was pretty significant.
However, Falkenhayn was unpopular in Germany army politics and had been on the verge of getting fired almost from the moment he took over at OhL, so I think the Rumania thing was just the excuse or the straw that broke the camel's back. My sense is that there would have been something else, though I can't think of anything on the top of my head. Another potential butterfly is that the 1916 Austro-Hungarian offensive against Italy was actually pretty successful for a World War I offensive, and maybe with less of an emergency with Galicia and Romania it would have been allowed to proceed and make more gains.
a) Yeah, to break with the trope of "kill everyone for a POD" that seems like a cool idea. There are more than enough dead end positions for Russian generals.
b) Shcherbachev indeed seems competent. However I really have doubts that his level of competence would necessarily translate into the innovations and success that Brussilov managed to achieve.
And in the Tsarist army we have to keep in mind that there is always politics involved. Even if Shcherbache would draw the same strategic and tactic conclusions, does he have the connections and backbone to insist on a "Shcherbachev Offensive" conducted in the "right way"? The other Tsarist Generals were against a general offensive against A-H and STAVKA tried to influence the way that Brussilovs offensive was conducted.
It might be entirely possible that STAVKA prevents the offensive from being successful by ordering large troop concentrations instead of a wider front.
c) If we assume that the main attack is happening in the West, there is a high possibility that the Southern Front has to sacrifice artillery, munition and men at some point. Evert and other traditionalists believed the reason for their failed offensives was a lack of heavy artillery.
If the political decision is made to mainly attack the Germans in the West, this will reduce the available ressources in the South. Just like IOTL Evert's offensive will quickly fail.
https://en.wikipedia.org/wiki/Baranovichi_Offensive something like this will be the result. This leaves the Russians two politically possible options:
1) Strenghten the West and keep feeding men and material into the grinder. The strategic rationale here would be breaking the Germans in combination with the Somme and Verdun.
2) Shift the offensive focus southward if Shcherbachev manages some modest intial gains.
With option 1) Baranovichi will become the "Eastern Verdun" but with even more loopsided casualties. Option a) would be the ideal outcome for the Germans. A-H i pressured much less and will be able to conduct offensive operations in the Ukraine. I doubt that the Germans would be forced to relocate as much troops as OTL to the East.
Option 2) is interesting. Ultimately A-H will have lesser casualties and will lose not as much ground. Even Brussilov himself stopped using his new approach towards the end of his offensive.
So instead of the phyrric victory with catastrophic losses we might see a tactical defeat for the Russians but with much lower casualties for both sides.
This currently leads me to the following scenario:
Shcherbachev, due to STAVKA meddling and not being a military genius like Brussilov, isn't able to implement the Brussilov-approach to the same degree. He also fails to enforce a strategic focus on A-H.
He manages to throw back the Austrians but his territorial gains are more modest. The Southern Offensive quickly peters out after some intial successes the Austrians manage to stabilize the situation with a little German help.
In the West Evert is forced to attack and quickly runs into the same problems that plagued him IOTL. Instead of ending the attacks ike IOTL, political pressure forces him to continue. He successfully lobbies for addtional reserves and material that is drawn from the Southern Front.
The Germans can shift their troops from the South to the West, without needing large scale redeployments from France.
Without tangible Russian gains Romania doesn't join the Entente and Russian losses are actually not quite as catastrophic as IOTL. So Russia actually is better off than IOTL because without Romania and with lower losses they still have enough strength left for another major offensive. Psychollogically the Russian situation is much worse though. Without a much needed victory, even the extremely phyrric one of OTL, there is a serious possibility that the Tsar is forced to seek peace earlier than IOTL. From the Russian perspective ITTL another offensive would be yet another waste of men and material.
Germany ITTL has more troops on the Western Front, so either the Entente faces more troops at the Somme or the Germans continue attacking Verdun longer than IOTL.