In this particular case I think it's more about the fact that it's Star Trek than the style of television or how serialized it is. Deciding that the recovery doesn't need to be shown is treating DSC on its own terms, as a show that takes place in the Star Trek universe. Asking how Tyler has recovered the next episode is kind of like asking how our heroes ended up on a planet thousands of kilometers away after a scene break - we know about transporters and we don't need to see them in use every time.
It's also understood that their ability to heal injuries is advanced, and they are treated differently than they would be in other shows. There have been many times where the drama isn't an injury itself but the fact that they're cut off from their ship (and therefore sickbay), and we know the resolution is reached as soon as they're able to beam back.
In context, Tyler's injury was never portrayed as "super important" because it was never implied that it was unusually serious relative to other injuries that they have no problem fixing in this universe. As long as it's believable that he could have returned to sickbay in time, I don't know why we need to give it a second thought.
That's a slippery slope: they didn't need to show any recovery because super tech --> they didn't need to show warp because warp is common --> they don't really NEED to show anything.
This isn't "Star Trek: the Franchise" or "Star Trek: the Setting", this is Star Trek: the Serialized Melodrama where things have (supposed) consequences. They show Burnham recovering because it's essential to the melodrama. They don't show Tyler recovering because his injury was written in a different episode and they don't feel like dealing with it because it's Burnham's story, which isn't a terrible reason at all.
That's not the hill I'm going to die on. I don't care if they show recoveries here or there, unless it's vital to the story: if they make a big point of the injury being plot/character important, then it should at least be mentioned. In this case... eh. 50/50. No big deal. It's more
indicative of the issues with DSC's structural writing than a problem in and of itself.
On the other hand, I might die on the hill of appearing/disappearing characters. These matter immensely to the serialized melodrama. Their nonsensical presence or absence absolutely matters, or at the very least begs questions that have no answers and reduces the impact of the intended stories because unlike a perfectly reasonable assumed offscreen recovery by
tech, the setting has no such outs for disappearing at warp or not showing up when relevant because the character is only in x number of episodes despite being just out of frame or in the next room or something.
That's what I mean by judging the show on its own terms. If we were judging it as Another Trek Show In the Usual Format, these don't matter and, in fact, are rendered moot because of the separation of time and plot. But DSC screams: "this isn't just Another Trek Show in the Usual Format, characters' arcs matter, things have consequences and long term repercussions, otherwise we wouldn't be serialized!"
