Brown might have thought that it was about protecting the public, if he thought that he was truly deluded. Brown was advised that bailing out the banks was the wrong thing to do, based on the Japanese crisis in the early nineties, which recieved a huge amount of QE into the financial system to protect it which it still hasnt truly recovered from, if you look at it in standard economic modelling terms. I think the Blair brown administration was far too heavily influenced by the City, when you look at the various schemes of the era, PPP, off balance sheet accounting, etc etc. and huge numbers of consultants within the labour administration from the big banks and big accountancy firms, so its no surprise that when the financial system blew up the banks would be protected. There is some writing that points to advance agreements on the banking bail outs going back a year or two prior to the crash, that the banks knew there was a strong possibility of a crash, but im not convinced it rings entirely true.
People will have the opinion that it was the right thing to do, but ive seen enough to demonstrate that it was probably the worse thing that Brown could have done, because the banks havent changed their behaviour one iota. The banking system could have been supported, either through being taken over by the central bank or nationalised (might not be much difference there) or even ring fencing the day to day banking system from the "investment" banking side even if temporary or some other means Richard Werner had a bunch of methods, but to reward the banking institutions and the people that were running them was insane and I think the outcomes were predictable.
The banks are now incentivised to operate in that exact same manner; banks when they fail next time, and it is going to happen, will have to be bailed out, but the problem would potentially be so big that there is a doubt that the banks could actually be bailed out, (because of asset footprint being so much bigger than GDP) so some other mechanism would have to be employed to make them whole again. One point of view I've seen is to detach currency creation from sovereign ownership (which banks are already doing anyway - which was a contributor to the crisis), and allow banks to create their own currencies. Some of the big banks are actively pursuing their own versions of crypto.