Superman #65 (March 1992)
Superman #65 (March 1992, Triangle number 10) - "Panic In The Sky: Second Strike" (part 3 of 8, prologue included) is arguably the first time the superhero community acknowledged Superman's character (and not just power or might) as exemplary and superior, specifically, in a leadership or sense of awe. Maybe. Although primarily driven by necessity, Superman assembles a huge team of heroes. The heroes also acknowledge Superman's leadership, plans, and protocols (Kryptonian code words). Guy Gardner begrudgingly admits Superman "is not a wussy", Ice is awed by Superman's looks, Batman actually follows orders, Elongated Man emits a "Woooow!", and Captain Marvel defers to Superman the entire time. Superman makes a number of speeches and they're welcomed home as heroes with a parade.
Even so, this interpretation and the event itself seems mostly a footnote in Superman lore. The in-story exaltation of Superman's character as beyond that of other superheroes seemed to start with 1993's "The Death of Superman" and beyond.
To see if this is the first, it's easier to start with exclusion.
Just Before The "Man of Steel" Reboot
Consistent with the question's thesis, Superman remained quite goofy right up and until his 1986 Man of Steel makeover by John Byrne. For context, during the same release prior of Frank Miller's "Batman: The Dark Knight Returns" from February 1986 to June 1986, in Action Comics issues 576 to 580, Superman would fight alien dinosaur men, stretch his cape to rescue a roller-coaster full of riders, travel to 253 A.D. with Jimmy to stop a fight between the Barbarians and Romans in Gaul, and transcribe his thoughts in a diary the size of a bank vault. In only a few more issues Alan Moore would do his deconstruction and farewell to the age in 583's "Whatever Happened To The Man of Tomorrow?" In sum, there still wasn't a consistent and grounded continuity to maintain any sense of moral superiority yet. That would come after the modern reboot. After that, to determine how other superheroes felt about Superman, we'd have to look occasions where other superheroes interacted with him, such as crossovers, event books, team-up, guest appearances, and the like.
In Action Comics
Conveniently, after Byrne's "Man of Steel", Action Comics 584 to 599 became a team-up title with Superman. 600 was an anniversary 80-page giant issue. 601 to 642 Action Comics became an anthology series. A quick review of these:
585 - Phantom Stranger does remark on Superman's character, however, you get the sense that he would wax poetic about anyone. Superman is frustrated with how obtuse the Stranger is and it's clear the Stranger's knowledge is preternatural (at one point Superman complains about him being in Kent's apartment), so his judgement of Superman's character is more supernatural than a common community-held belief based on Superman's visible exploits.
593 - Superman is called "pure and noble" by the villain, who is trying to get Superman to star in an adult film with Big Barda, which puts into question his threshold for nobility.
Generally, he's an ally, but none of the other heroes put Superman on a pedestal because of his morality. Robin is star struck in 594, but that is more about Superman's feats and celebrity. Similarly, in 600, Wonder Woman ascribes him "might equal to the gods" in terms of power, not character.
601-642 - During the anthology period, Superman's story involved a cult that worshiped him as a god in an absolute sense (he's good because divinity is good, not because of who he is or what he does). Hal Jordan's stories occasionally gave insight into how he thought of Superman... not in awe or reverence, but consider Superman somewhat obnoxious.
Superman returned to the tile in 643 (July 1989). For context, Adventures of Superman 463 (February 1990) featured the race between Superman and Wally West, with Batman actively disrespecting Superman on the cover and Wally showing Superman little respect in the issue itself.
650 (February 1990) - the Justice League International shows Superman no respect initially, J'onn chides, "To belittle Superman in order to build yourself up is unnecessary and unbecoming a member of Justice League International!" J'onn tries to share how Superman first helped the League but declined membership and Fire accuses Superman of being "a snob" who "sounds to me like he was really full of himself!". J'onn and Powergirl try to rehabilitate Superman with additional feats, but the conversation veers off.
654 - The iconic gift of Kryptonite from Superman to Batman which begins with, "I'm sorry, I'm not in the habit of barging in on friends unannounced, but then, we're not exactly friends... are we?" Batman is effectively mute throughout the scene so there's little indication on how he feels about Superman.
674 - Panic In The Sky starts.
In Justice League
After "Crisis on Infinite Earths", the Trinity (Superman, Batman, and Wonder Woman) were no longer founding members of the Justice League. In their first adventure, Wonder Woman was replaced by Black Canary, Batman, no longer appears, and Superman appears at the end to help, but declines membership.
After 1986, Superman is not a part of the Justice League books until Justice League Spectacular #1 (April 1992), after "Panic In The Sky: Second Strike."
In Crossover Events
Between 1986 and 1993's "The Death of Superman" event, I can't point to an event where Superman's character was recognized as exemplary by other superheroes.
While Superman has always been an aspirational figure to his audience in terms of powers and general heroism, and an inspiration- creatively- spawning an entire creative genre and industry. He has not served as spiritual inspiration or a moral pillar above and beyond his superhero contemporaries within continuity and within his story for very long at all... roughly starting in 1993. It's difficult to say when the period ended (if it ever did), since fandom would continually complain about his feet of clay from then on as well, and that viewpoint would make it into the comics and continuity.
Famously, in Infinite Crisis #1 (December 2005), Batman would say, "Everyone looks up to you. They listen to you. If you tell them to fight, they'll fight. But they need to be inspired. And let's face it, 'Superman' ... the last time you really inspired anyone was when you were dead."
If a reliable statement (it isn't), in continuity, the first and last time Superman was a moral inspiration to look up to and listen to was more or less the same brief period.