But having allies, especially Bat-themed allies, adds to Batman, not detracts. One, it makes it all the more believable that Batman could even have the notion that he could save Gotham City, by training more people to aid in that effort. Two, it plays into the idea of Batman always being prepared. Having multiple allies is his insurance policy if he is ever killed or even when he eventually dies. Why? Because, in this way, he knows that there will always be someone defending and protecting Gotham. How is it that we can be against the Bat Family showing up yet in support of an anti-hero like Catwoman showing up? It makes sense for Batman to have allies who are fighting alongside him and under his tutelage/training. Third, there's absolutely nothing wrong with other Bat-themed heroes. Contrarily, having Bat-themed heroes reinforces the idea that Batman has inspired others. As for how this could fit with a Batman v Superman where Batman is alone in Gotham? Simple. Batman has pushed his allies away in the wake of Jason Todd's death and a deteriorating situation in Gotham, in part because he wants no one else close to him to be hurt or killed. Batman's journey out of this state of mind (which is all but guaranteed to happen by the end of Batman v Superman) can result in him reconnecting with his former allies. Not only that, but I wouldn't understand a Batman that is willing to fight alongside the other members of the Justice League yet isn't willing to have allies in Gotham. As for Oracle, there's no point having her if she is never Batgirl. I'm sorry, but it's pointless imo.
I can understand and get on board with the idea of Batman inspiring others to do the same he does, though if we're to talk about what is believable, half of them, if not more, should be dead already. It should be established that not everyone can do something like this, again, Batman is an extraordinary human being.
And while you make a fine point with it going along with Batman's tendency to have contingency plan, in case he dies, there is a problem of Batman usually being depicted as rather territorial, even in arcs where there's ton of other crime fighters such is "War Games" and writers often go against themselves with that - if he is so territorial why the hell did he let so many other costumed crime fighters run rampage through the Gotham?
Also, him not planning for someone to take his mantle might be explained with two good reasons:
1. He wouldn't want for anyone to have a life such as his, knowing at first hand how hard and dark such a life is.
2. He might feel arrogantly self-confident enough to never consider being defeated while Gotham still needs him, hence needing someone to continue his fight. It's in character of him to be stubborn like this. This is a man that refuses to see that he is fighting against the windmills and how pointless his ongoing crusade actually is in a long-term, after all.
As for other bat-like characters, it is a matter of aesthetic connection, a symbolism if you will - when you see a bat-like bipedal creature in the night in Gotham, that is Batman. Making it, perhaps it's Batman, perhaps some of the many kids he inspired to dress in a similar manner waters it down to me.
And Catwoman serves quite a different purpose than Batman's direct allies. She is a standalone character. She doesn't have a direct connection with him such aforementioned allies do, she is almost always depicted as morally grey with her own agenda and while she might help him out occasionally, she would never take orders from Batman. She is a character in the same vein as any from Batman's rogue gallery, it's just that she falls in the middle of the spectrum and from time to time allies herself with him.
Also, I don't see why would it be pointless having Barbara Gordon that is an ally to Batman and uses her computer knowledge to help him, if she was never Batgirl. Her previously being Batgirl adds nothing in that regard, especially since she was shot by Joker because she was Gordon's daughter and not because she was fighting alongside Batman in a costume - Joker didn't know that it was her when he shot her.