Except no, because all PaulGreenie would have likely done is transition in and burn their heart out. Even if Batman catches on to the admittedly unusual murders spread all over the states and starts investigating, there isn't any more evidence than there was to start with - he'd still have to investigate and work his way up the chain.
Paul was several steps ahead, knew who he was going after, and had Power Ring-derived hacking and tracking abilities, so while I'm not saying Batman's unable to match that effort - in fact, I'm noting the opposite - that Bruce went from zero to finish faster than Paul can only suggest his previous efforts were not so dedicated.
Paul mentioned that the intelligence services had likely figured out who it were that he was killing, so quite likely did Batman. And even just finding out names he didn't know before allows Batman to search for connections between them to track assets.
Also, it would stand to reason that some of those databases Paul found were near the Shadows he killed. There's nothing saying Paul destroyed them after, so Batman might have found them following the bodies and hacked them too.
Also, nothing says that Batman started from
zero, he just didn't know enough to directly pinpoint to Ra's. Following after Paul who was using methods Bruce would be unable and unwilling to use might have given Batman the last pieces he needed and, being a lot better detective, made the connections before Paul did.
To put their roles in somewhat-less fantastical terms, if I knew that an officer of the law was a lot more dedicated to or talented at saving the lives of serial killers than actually stopping them - especially when the skills to do so are the same - then I would have some serious concerns about the motivations of that officer and the suitability of his service.
It would be more accurate to say that Batman is more dedicated to stopping murderous vigilantes, than murderous supervillains.
Batman is afraid he would snap if he killed someone. He might feel, that if he didn't stop someone, who he actually
wouldn't mind dying, from getting murdered, that he is also responsible, because he might have held back subconsciously, to let someone else do his dirty work. So he gives his 110% into stopping it.
And yeah, it varies depending on the work, but most of the time it's basically canon that he is almost as damaged as his rogues gallery in many ways, he has just found a more moral and constructive way to channel it. But yeah, no realistic police force would hire Bruce, after taking one look at his psych profile.
