When it came down to the decision for Naruto, it was a small panel of evaluators. One was Iruka--a close acquaintance of Naruto's--and another guy was for passing him (so he could manipulate Naruto to his own ends). The deciding vote was Iruka's alone, and it was Iruka that specifically insisted that Naruto must be able to create a plausible clone. The reason Naruto failed in that specific case is because of one particular man's expectations alone (to be fair, he had a pretty good reason). The general hatred the village had towards him, even despite the Third's orders, would likely have worked against him more generally. Lee would not have had that same hatred.
POSSIBLE SPOILER: The following makes use of some of the background of Might Guy that we only find out very late in the story.
The histories of Might Guy and Might's father give precedent to people exceptional at taijutsu being acceptable based on that alone. Exactly why is never really stated. We know that Might had troubles entering the school due to his poor non-taijutsu skills. Possibilities for advancement in these cases include:
Training so hard and demonstrating such exceptional skill that the village had essentially no choice but to recognize him. He, possibly like Lee, may have simply "skipped" the usual method to becoming a Genin.
A special case of the first point is that the ability to use the Gates is a rare, risky, and potentially extremely powerful skill. Possession of such a thing may be sufficient reason alone to advance in ninja ranks. This may both be as a precaution--make sure he knows how to use it lest we suffer for it--, or as a simple recognition of its usefulness when properly utilized. Naruto was not expected to have any control over the beast by default (only the upper echelons seemed to be aware of the possibility), which is in part why they feared him. The beast was only seen as a possible source of death and destruction, nothing more.
Might's father had an (entirely unexpected) epic last stand by virtue of taijutsu. Possibly word got out, and this led to a substantial improvement in the perception of taijutsu users. This may have created a special, taijutsu specific, path of advancement that was now available to Lee and Might.
Related to the previous: Advancement tracks for single-area specialists were already in place. Naruto was not a specialist in any specific thing. He was training under a more general program, which requires a broad range of skills but at less depth. Lee may have been under a specialist's program, which requires a narrow range of skills but at greater depth.
Might may have already had his eye on Lee, and may have used his stature and influence to push Lee through and into his team. We still need to consider a reason for how Might got his stature.
- Note that, out of universe, Might was initially never suggested to be only good at taijutsu, and it was initially clear that he could use chakra with reasonable proficiency. We see him using summoning skills (for turtles) and breaking genjutsu on his own up through the Chuunin exam. It was really only after this exam that his non-taijutsu skills basically vanished, and Might was largely recast as being very similar to Lee. So Might being a high ranking ninja may have been nothing more than a continuity requirement: we already knew he was, so he had to stay that way, even if he no longer met the prerequisites. So the door for Lee was opened by a silent retcon, and no in-universe explanation for that door was provided (lest it draw too much attention to the retcon).