this is a little tl;dr, but... From then on, he worked alone.
But people keep wanting to bring back the brightly colored children, and the lighthearted camp silliness.
I believe Tim Drake was introduced only a couple years after the death of Jason Todd, even if he didn't don the Robin costume right away.
Certainly the two versions of Batman you describe–one being the dark, grim, and a slightly gothic and romantic hero who is alone in his unrelenting mission, and the other being completely light-hearted and campy, with silly gadgets and in ridiculous scenarios–both exist and are both a part of Batman's history, and are always going to inform the character.
But to me, Robin, and all the many Robins and other younger heroes that gravitate to Batman and become a part of his story, allow for a greater depth to that dark and grim loner hero. It allows him to be more multi-faceted. He doesn't just seek justice and protect the innocent in his "work life", he also extends that compassion and the desire to right wrongs into his personal life.
He takes in orphaned Dick Grayson for the same reason he became Batman in the first place: because he doesn't want anyone to experience the pain he did. He doesn't want people to die at the hands of murderers, but that's what happened to Dick's parents. And knowing firsthand the pain that Dick would be going through, knowing better than most people what will help and what won't, he feels compelled to take care of him and be there for him.
I will grant you that having Dick turn into a crimefighting partner at such a young age is where things start to get problematic (child endangerment, why wouldn't you make the kid wear pants? etc), but add of bit of superhero logic and it's workable. Dick wanted to fight for the same things and for the same reasons that Bruce did. And mentoring him to do the same allows them to have a better relationship IMO, because of their shared goals and all the time spent together working toward them (if Dick wasn't Robin, Bruce would probably rarely ever have time to see him). Although Bruce ends up teaching a lot to Dick, Dick in turn teaches Batman.
He's a reminder that not everything is bad, that people can recover from bad things, and most importantly that he is not alone. All of Batman's darkness and the loneliness of his mission can easily get depressing and become just as ridiculous and unrealistic as shark-repellent spray if it's not checked once in a while. Robin is that check. (I would also argue that Robin can also function as a way of pointing out and making fun of how absurd Batman all by himself in his original dark lonely mission, but taking it over the top with his circus-coloured costume.)
With the exception of maybe Jason Todd's introductions, all the other sidekicks/younger heroes that revolve around Batman weren't introduced in a way that's campy or all that silly. But collectively, when you look back on the entire history of Batman and his young pals, it is a little ridiculous. There have been five Robins now, two Batgirls, Spoiler, Huntress, there's Alfred, and he's also mentored Blue Beelte III a bit (and in the DCAU you've also got Terry). Dude's got a big family and he's mentored a lot of kids, some of whom are his children by blood, adoption or with home he has close relationships.
I think what Black Cat's doing here, and why many people enjoy it, is that it's acknowledging some of Batman's accumulated family, acknowledging the absurdity but also the warmth and familiarity of it all (and if Bruce wasn't such a loving person, I can't see him having so many friends and kids hanging around him). And considering that the family-life of Batman is not usually the focus of canon (which makes it all the more special and loved on the occasions when it does), this kind of fan comic is a bit refreshing and fun because that family dynamic is the focus, rather than the sideplot.
Basically, when it comes to canon, I don't think the only options are brightly-coloured campy silliness and grim, gritty, dark, gothic loneliness or that you can only have one or the other. It's not such a sharp dichotomy: there is a lot of middle-ground that can be explored, and probably my favourite versions of Batman incorporate elements from it all.