My Hero Academia isn't just the story of how Izuku Miidoriya becomes the world's greatest hero, it's also the story of how Tomura Shigaraki becomes the world's greatest villain. I believe that All For One will someday transfer his quirk to Shigaraki.
Izuku Midoriya was quirkless and Tenko Shimura was believed to be quirkless; but both dreamed of becoming heroes in spite of this. There is also another parallel between their origins.
In chapter 1/episode 1 In chapter 1/episode 1 of the series we see Katsuki Bakugo explode Izuku's desk then, Izuku scampers into the corner and Bakugo stands over him yelling. All this is done in full view of the teacher and the other students and nobody helps Izuku. This moment implies that the torment Izuku suffered from Bakugo was enabled by teachers refusing to punish Bakugo.
Tenko Shimura grew up with an abusive father who would lock him outside, even for the night, and sometimes beat him just for mentioning heroes. Tenko's mother, sister, and grandparents all knew what was happening but they didn't save him. Bear in mind that Tenko was only about 5 years old when these events happened to him.
Tenko Shimura was found by All for One, the world's greatest supervillain and raised to be his successor. While Izuku Midoriya found All Might and was chosen to become his successor.
The mirroring of Izuku and Tomura has been going on since the USJ Incident. At this point Izuku Midoriya and Tomura Shigaraki are both nerdy amateurs, with "awesomeness by analysis" and an obvious tic. Izuku has a verbal tic, his mumbling, and Tomura's tic is scratching his own neck.
This arc contains two moments of Izuku rushing to save All Might, the second at the same time Tomura is rushing at all Might to kill him. Despite their efforts and intentions neither Izuku nor Tomura succeeds at their goal. Izuku Midoriya is blocked from saving All Might both times by Kurogiri, and Tomura Shigaraki is stopped by the arrival of the teachers.
That's not to say that Deku and Tomura were passive or pointless during this arc. Izuku succeeded at saving himself, Tsuyu, and Mineta and defeating the goons that were attacking them; Tomura did fight and succeeded in injuring Eraserhead.
As mentioned before Deku and Tomura are both amateurs during this arc. To begin with Tomura didn't plan this attack out particularly well and favored quantity over quality, assembling a small army of goons who were defeated by students with barely any training. The students at least seemed to struggle and the goons did manage to recover, but when the teachers arrived the goons were wiped out quickly and easily.
During Izuku's second attempt at saving All Might, Deku breaks both his legs propelling himself forward in an attack. I think this was meant to show how Izuku's self-sacrificial style of combat is impractical in a real fight against villains; something Eraserhead warned Deku about during the Quirk Apprehension Test.
Before each time Izuku rushes in to save All Might we get a scene that showcases Tomura Shigaraki's villainous flaws. When Nomu and Kurogiri had All Might restrained, if Tomura had given his enemy the touch of death he could have ended the symbol of peace. Instead Tomura relied on Kurogiri to close his portal, cutting All Might in half, and stopped to explain this to all might instead of just quickly ordering Kurogiri to close the warp gate. After All Might defeated Nomu and was about to revert to his true form, Tomura was willing to run away until Kurogiri convinced him that All Might was bluffing and that they could still win.
So Tomura is charging at all might to kill him and Deku charging in to stop Shigaraki. Deku breaks both his legs propelling himself forward in an attack. I think this was meant to show how Izuku's self-sacrificial style of combat is impractical in a real fight against villains; something Eraserhead warned Deku about during the Quirk Apprehension Test.
After Tomura Shigaraki is shot, Kurogiri envelops him and they leave, but as Tomura retreats into the darkness he tells All Might the next time they meet he'll kill him. When Shigaraki returns to the hideout we hear the voice of All for One for the first time and he refers to Shigaraki as a "symbol" hinting at his larger significance in the story.
It seems counter-intuitive to introduce your final villain this early into what's hopefully going to be a long-running Shounen Action series, and especially to do so in an arc that emphasizes all his weaknesses instead of building him up as this big intimidating obstacle that the hero must work tirelessly to catch up to. At this point in the story Tomura Shigaraki is little more than a Saturday morning cartoon villain who shows up to cause trouble and is defeated and chased away promising he'll get them next time. Similar to Izuku's introduction as a powerless boy who still leapt in to save Bakugo, this introduction to Shigaraki gives a greater appreciation to how much he grows and evolves over the story.
The Hero Killer arc gives both the hero and the villain a chance to address one of their greatest weaknesses with the help of a temporary and brief mentor.
For Izuku Midoriya the weakness is his inability to properly control One for All; during the Sports Festival he pushed his self-destructive combat style to it's limit and now he knew he needed to learn a better way. With the help of Gran Torino, Deku overcomes this weakness by developing Full Cowl. For Tomura Shigaraki the weakness is his lack of conviction. Stain was brought in, presumably on the orders of All for One, to help teach Shigaraki. Shigaraki doesn't listen to Stain however and it's only at the end of the arc that he recognizes that Stain has something he doesn't.