What did the Senate actually do in the Byzantine Empire?

First, it's good you're asking about the role of the Senate in the sixth century, because for anything after that, it's parsing through a bunch of vague references to try to paint a picture of an institution that was evidently given little to no consideration by any actual Byzantine power player. While the Byzantine Empire hardly experienced a thousand years of decline, the Senate's importance certainly did.

Your assumptions are essentially correct. The Senate, or in Greek the synkletos, was composed of men of class and rank that would have been included in the Roman Senate, and in fact many of the Senators were descendants of the old Roman aristocracy. There were also a number of people in the Senate that the Romans would have considered nouveau riche, but in the primary sense, every single Senator was either a great landowner with extensive holdings across the Empire or some other important figure in government (the two often went hand in hand). Thus, the Senators' authority came more from the fact that they were each important stakeholders rather than that being included in the body had any authority in and of itself.

Following from the idea that Senators were such because of their importance in government, the Senate had some measure of organization in that its members were classified into three groups: the illustres, spectabiles, and clarissimi (I will refer to them as first, second, and third rank Senators respectively). The first rank Senators were the only ones that regularly actively met in the Senate houses, and by Justinian's administration, they were the only Senators permitted to meet at all. They held the highest offices of government, such as magister officiorum (head of the civilian administration) or magister millitum (head of the military administration for a particular frontline). All these positions were based in the capital itself, and helps explain why they were the only Senators that met. The second rank was composed of the various mid-level officials, such as vicars and proconsuls, of the Empire, as much as those titles actually still predicted their functions. The third rank were initially the lower provincial officials, though Justinian began/expanded the practice of selling the title of Senator to anyone who could buy it. The third rank was thus both those lower officials and people who could buy into it.

While an Emperor reigned, the Senate acted as a purely advisory body that could help create legislation- if the Emperor consented. The body regularly met, would often read out new laws before their enactment, and could pass resolutions regarding the administration. If the Emperor so desired, he could turn those resolutions into decrees with the force of law, though he was under no obligation to do so. The Senate also possessed a nominal judiciary function, in that it could act as the highest court of law in the Empire- though it was essentially never called in for this duty. The princeps of the Senate was no longer the Emperor, but rather the prefect, or eparch, of the City of Constantinople. In contrast to the Senate as a body, the Eparch of the City had teeth, and was included among the first rank of members in the government, and this would continue long after Justinian's death.^(1)

Where the Senate actually had some measure of its own influence was in regards to succession. When the Emperor died, the new Emperor had to, at least in theory, be acclaimed by the Senate and Army. If the Emperor had designated a clear successor, this was little more than a formality, if not, the Senate played a role in deciding the next Emperor. It should be noted this scenario was not entirely common- in two hundred years it only happened three times, and none of those occurrences put the Senate as a body in a decisive role, though individual members were quite influential. The first followed the death of Marcian, the last of the Theodosian dynasty in 457. At this point, the government was effectively controlled by Aspar, a Germanic(?) general who had positioned his influence among the foederati to become indispensable (read: irremovable without risking a very serious revolt among the German armies). There were no Theodosians left, so it fell to the Senate to decide the succession. It took them ten weeks, after which point they (wisely) deferred to Aspar, who chose an officer of his named Leo.^(2)

The second occurrence actually brought Justin to power. Anastasius died without a clear heir, so the throne quite literally went to the highest bidder. The Senate was assembled to elect the next Emperor, and Justin, a mere bodyguard with control of the crackiest of the crack military units he and the other candidates had, was apparently able to bribe the right Senators. The third followed Justinian's death. Justinian neglected to name a clear successor, so again the Empire was up for grabs. In this case, Justinian's nephew, also named Justin, had spent much of his adult life making allies among the more important stakeholders in government. The Senate, composed of those stakeholders, was thus made partial to Justin's accession, and a delegation of the Senate actually went to Justin and offered him the Imperium without much other trouble.

So, as for the Senate's importance here? Mixed bag. As I indicated above, most of the Senate's importance derived from the fact that the Senators were each individually important. Decisive roles, in anything really, fell to individual Senators rather than the body, in contrast to the case three to four centuries prior. Overall, you do get the impression that the Romans were keeping the body for formality's sake. Over time, selling the title of Senator would become more and more common, and references to it become more and more scarce. The Basilisska, a revision of Justinian's Corpus Juris Civilis, would effectively discontinue the remaining functions of the Senate, though it would continue to nominally exist as a body until it dropped off the radar entirely in the fourteenth century.^(3)

Constantinople was one of the largest cities in Europe for the duration of its tenure as the Roman capital, and was the center of all the Empire's trade and manufacturing, a role that was only magnified after the losses of the seventh century. Managing the guilds, tariffs, and other associated nuances of a city of, at times, more than half a million people was a job in and of itself, even more so when one considers how heavily the Emperor's personal finances were invested into trade and manufacturing within the city. As you can probably imagine, the person chosen to be Eparch of the City would given the additional responsibility of leadership of the Senate, rather than the leader of the Senate being given the additional responsibility of Eparch.
It should be noted that Marcian was already a product of Aspar's kingmaking, but in that circumstance, a Theodosian scion, Pulcheria, was still alive, and deriving that much needed legitimacy to rule was as easy as coercing her into marriage. Pulcheria died in 453, and was thus unavailable to marry after Marcian died a few years later. Alas!
Even so, it's not like it maintained importance during that time period. Anna Kommene makes no direct mention of it in the Alexiad, and the last action it may have helped undertake was during the chaos of the Fourth Crusade, following Alexios IV's deposition. A man named Nikolaos Kanabos was acclaimed Emperor by a collection of the more important members of the cities, including what Choniates refers to as the "synkletos", but if you want an indication of what Kanabos thought of the legitimacy of that, he refused to accept the nomination and locked himself in the Hagia Sophia until Alexios V, a candidate with military backing, dragged him out and strangled him.

Sources

Alexiad, Anna Komnene, trans. E. R. A. Sewter, revs. Peter Frankopan, 2009

Dementia on the Byzantine throne (ad 330–1453), Geriatrics & Gerontology International, Effie Poulakou-Rebelakou et al, 2011

Emperor and Priest: The Imperial Office in Byzantium, Gilbert Dagron, pp. 320-326

"Eparch of the City", The Oxford Dictionary of Byzantium, Alexander Khazdan, 1991

History of the Byzantine State, George Ostrogorsky, 1969, pp. 27-50, 245, 324

O City of Byzantium: Annals of Niketas Choniates, trans. Harry J. Magoulias, 1984

Note: Blech. It's been almost two years since I've had access to my university's online library sources, and my computer with all the articles and documents I saved on Zotero died while I was living abroad :'( Relying almost entirely on print sources and google searches is awful, and I think some of the things I cited reflect that lol. Still, I hope this was useful OP. At the very least you gave me something fun to do for a couple hours haha.