In the manga they mention that they get recommendations based on the students performance in the tournament. Although Midoriya performs really well there and manages to reach the top 8 he doesn't get any recommendation(apart from the single one) why is that so? He wins the first round without using his quirk and although his quirk harms him, the quirk in itself was exceptional if brought under control. So was curious as to what kept the pro heroes away from him.
You yourself mentioned the main reason. IF brought under control his quirk is exceptional. However, we as readers get the extra information that the characters of the manga don't have.
He inherits the quirk from All Might.
Thus, Midoriya maybe capable of big things, but the other pro heroes will consider him a liability in practice because an injured person removes 2 persons from a fight.
Also note, though he achieved good results, he didn't do anything impressive showcasing his hero talent.
Obstacle Course Race. He won the first round because Bakugou and Todoroki were fighting against each other. 1B wasn't interested in the first place. He didn't use his ability at all! This is mentioned specifically in Manga.
Cavalry Battle. They barely scrape through. Though Midoriya planned well again. From the outside the MVP for the team was Tokoyami who defended well and was able to scrape through the qualification in the end.
PvP Tournament. 1st Round. Even after knowing his opponents weakness, he gave himself away. He lucked out in the end, when Previous One for All users triggered him from the hypnosis. He won since his opponent was weaker when they both didn't use quirks.
PvP Tournament. 2nd Round. Lost to Todoroki. He got his ass kicked tbh. He showcased his will power and helped Todokori assess his powers, but he was literally helpless. From the outside, it showed other Pro Heroes his vulnerabilities rather than his strengths!
Now, its different. He has formed his own style of battling and is interning with AllMight's former sidekick in the current arc! So look forward to Anime/Manga. Cheers