In the final scene, when Saru and Burnham are looking out the viewpoint they are able to see straight across to Starfleet headquarters. However, the subsequent external view shows that the view from their port is blocked by the outer ring of Discovery's saucer section.
In order to steady the ship in the ion storm so that a tractor beam can be applied to the Tikhov, Detmer says she is engaging "reverse thrusters." Reverse thrusters, as the name implies, cause a starship to move backwards, which is not what Discovery was trying to do at the time. Pulses from the ion storm were already forcing the ship back. Detmer should have been using maneuvering thrusters to counter the ion pulses' force upon the ship.
While marveling over the new Starfleet ships, Owosekun is able to identify the new Constitution-class ship. Since the 32nd-century Constitution class looks nothing like the 23rd-century Constitution class they would be familiar with, certainly no more so than any other Starfleet ship, this makes no sense.
Saru's description of the "dark" Middle Ages corresponds more to a cliche than reality and already no longer corresponds to scholarly opinion. Giotto was also not someone who changed the view of the world in a single achievement, but rather built on the achievements of his teacher Cimabue, who overcame the Byzantine formalism and medieval art that had prevailed up to that point. Without the Humanism that accompanied the Renaissance, starting with Petrarch and able to build on Dante, the new world view would not have come about either. Of greater importance than the change in perspective by the artists was the new worldview in which humans became the focus. Nevertheless, wars did not become fewer, on the contrary.
The medical officer reporting to Admiral Vance says that her patients are being affected by "something called prions." Prion diseases are well-known to 21st-century medicine. Certainly doctors in the 32nd century would have even more knowledge.
The 32nd-century holographic doctor, Eli, is somehow much less natural in its mannerisms than the holographic doctors seen aboard Starfleet ships in the 24th century.
Admiral Vance says that, because Saru and Burnham's story contradicts the historical records about Discovery's fate, there's no way for him to corroborate their version of events. That's what logs and sensor data are for.
Entering the seed vault requires Burnham to call over to Discovery and have them use their transporters to put her inside the vault. If they were capable of that, there is no reason the away team should have had to have beamed to another part of the ship and walked to the vault.
In his opening log entry, Saru describes the joint Federation/Starfleet headquarters as "separate entities that must no co-exist." While it's true that Starfleet and the Federation are not synonymous, they were never wholly separate, either. Starfleet is an exploratory and defense organization that exists within the purview of the United Federation of Planets.
Michael tells Admiral Vance that, had Discovery not made the jump to the future, all organic life in the galaxy would have been wiped out. Previously it was said that Control was going to wipe out all sentient life, not all organic life. While the sentience of non-human animals is often debated, the organic nature of plants, fungi, single-celled organisms, etc., is not.
Burnham asks how it is possible that the Barzan family in the hologram aboard the seed ship could be humming the same melody that Adira played on her cello after absorbing the memories of Tal's previous hosts. As Federation member worlds often engage in cultural exchanges, it would not be at all unusual for something as simple as a song to be passed from one planet to another.
Admiral Vance states that it is his understanding that Owosekun and Detmer piloted Discovery into an ion storm to retrieve the drifting seed ship. Detmer piloted Discovery. Owosekun was simply in the seat next to her while doing so. Rhys was also instrumental in rescuing the ship, operating the tractor beam, but gets no mention.
During his opening log entry, Saru states that he hopes Starfleet is as eager for them to arrive as they are to reach Starfleet. Since Starfleet does not know of the existence of Discovery, there is no way they would be eager for their arrival.