Our knowledge on sidekicks is admittedly limited, but the information we do have, it doesn't have great implications.
From what can be gathered: somebody is sidekick if they work for another Hero's agency, regardless of their age, achievements or experience. Considering Deku, in his inner monologue, often speaks akin to "This is like a comic book" - It implies the Superhero genre existed in the MHA world's media beforehand. So when people started getting quirks, the public wanted to bring those comic fantasies to life.
However in traditional comic books, sidekicks are young people training under a mentor. So wouldn't it be a demeaning title to call an older professional this? Sidekicks could easily be older than the Heroes who run their agencies. For example, Centipeder was a Sir Nighteye's sidekick at 35. Since many start their hero careers after High School, he likely has 10 years experience at minimum. Yet he only became a Pro Hero due to Sir Knighteye's death and his agency needing to be taken over.
Hero agencies are overly dependent on their bosses compared to most businesses. Agencies are often known exclusively by bosses' names, and not by umbrellas titles like other businesses like "Google", "Sony" etc.
It can get to the point, sidekicks' uniforms are based round the bosses' personal fashion sense. Best Jeanist's sidekicks come to mind for this. Instead of a unifying uniform - The uniforms copy and bolster one individual, effecting their egos in unhealthy ways. Its borderline worship. With the hyper focus on the bosses, they are likely given too much credit by the media and their sidekicks' hard work are undermined.
It's implied to be judged seriously an individual hero, they need to own their agency. However simply being able to run an independent business/agency, doesn't say much about someone's true skill as a hero true. Their main job should be to rescue civilians or stopping villains, not to be the leader of an agency. It's not like Policemen have to found new stations to prove their independence.
Yet despite this, if a hero struggles to run a business, they'd stuck being someone else's "sidekick", their other skills and achievements overshadowed. Which I imagine causes widespread frustration for many heroes.
This pressure to create their own agency, I'd think could cause an economical issue. As in the oversaturation of agencies and hero profession as a whole. In the first episode of the Anime, the entirety of Deku's middle school class wanted to become heroes.
In that same scene, Bakugo even insulted them by saying "Those guys will be lucky to end up as sidesicks to some busted B lister.". Clearly spelling out, that sidekicks are seen as lesser, despite the only qualification to be one, is being a hero who doesn't own their agency.
Considering the major villains are based around specific flaws within Hero society, I think the potential issues around Sidekicks and Agencies could be explored. It would be interesting if we have a villain motived by being an disgruntle sidekick.