This is expanding on a theory from another thread in response to a comment about how unrealistic it seems at first blush for Boothby to know everyone.

There's actually quite a bit that's irregular about Boothby. There's no real need for human gardeners. The Federation has AI on its starships that can create natural environments from scratch based on verbal descriptions, and has mass transporter technology that can move things down to a molecular level. It also has tractor beams, force fields, and robotics. Not to mention a continual supply of military cadets in peak physical condition who can't say "no" to grunt work. Starfleet Academy doesn't need to landscape by having an old man dig around in the dirt with a shovel. And surely if they at all wanted to, they could replicate him something a bit nicer and more durable than a raggedy pair of patched-up overalls.

I'd posit that Boothby works at the Academy to provide a "how does this person treat service people" test. He's not there for his gardening skills, but because he's an extremely wise and affable fellow with an eidetic memory. He's uniquely capable of being totally innocuous, while being quietly brilliant at understanding people.

The natural-born leaders, the one that Starfleet is really interested in, they respond to Boothby as an equal without really thinking about it. They apologize if they get in his way, they instinctively smile at him when they see him in the morning, they start up a conversation with him or even offer to help for a bit. They're aware of the morale of the people around them, and they try to lift it without even thinking about it, whether those people are immediately important or useful to them or not. They build a community wherever they go. Those are the people that are likely destined for the Command track to become a Captain someday, who can reflexively work seamlessly with hundreds of people underneath them, many of which they may barely know.

Then there are those who acknowledge his presence, and are polite to him when they encounter him, but don't build that connection. They're more focused on their friends and peers and studies, and impressing their instructors. The clever and curious ones are intrigued by why Starfleet employs such an anachronistic method of gardening, but their conversation with him serves mainly to satisfy their own curiosity rather than to build a connection with him. Those people are the ones that may be destined for leadership of a small flat science team where they know everyone, but might struggle in command of a full starship or space station.

And then there are those who are outright brusque or rude, who are doing what they have to in order to put on the right face for the people that they think matter, but intrinsically don't care about people whose way of life is different than their own. They applied for the Academy for the status and glory. They rank themselves higher than Boothby and don't have time for him, or outright make condescending remarks about him to their friends as they walk by. Those are the people that Academy leadership takes a closer look at as to whether they belong at the Academy in the first place.

Boothby also sees the cadets when they go out into the grounds to be by themselves, when they're upset or overwhelmed. He always seems to be out there, so people pay him no mind. But he can gently start a conversation, and the most perceptive ones quickly recognize that his wisdom far outstrips that of gardening. They begin to seek him out. Their relationship with him helps determine how well they can have a confidante relationship with someone technically beneath them - an absolutely critical skill to have when they may be the highest-ranked person on a starship and need to ask for help to tend to their own needs.

There's no formal score given or reports filed. But every now and then one of the deans stops by and has a casual chit-chat with Boothby.

“Yes, it was quite an enjoyable morning. I met Mr. Crusher today. Quite a friendly young man. Stopped by while I was planting more flowers and seemed genuinely interested in the conversation. His companion though, a Mr. Locarno...he seemed to be in a bit of a hurry and getting impatient. Seemed to have much more important things on his mind.”

Hence why you get the successful people going back to Boothby, because once they get out of the Academy they’re clued into the open secret that Boothby was a soft test of whether they’d value all members of their crew, and not just the highly visible “important” ones. When they report for their first assignment, their superior officer may notice their profile is already flagged for the command track.

"I take it you know Boothby?"

"The...groundskeeper? Yes, how did you know?"

"Your file has you already flagged for the command track. That only happens if Boothby likes you."

This is also why Species 8472 has Boothby practically running the whole place. When they assimilated the information they had, they didn't have any context for it. They didn't understand that military ranks would most visibly dominate the social hierarchy. Instead, what they saw was that the instructors universally respected Boothby's opinions. If he said to give someone a second chance, they did. They also saw he was also indirectly mentioned by many successful Captains when they described the most valuable parts of their experience at the Academy. Their simulation recreated him based on the level of influence they believed he had, rather than his nominal rank.