What we need is success on the pitch, that is what drives our brand recognition, nothing else. Everton FC is bigger than the City of Liverpool now, we have the globalisation of the Premier league to thank for that.
The only reason the liver bird is recognised by anyone outside of Liverpool is the success that lot had in the last 50 years, it is associated with them, not with LCC or the symbol of our city. 
I left Liverpool a long time ago when I was 16, I have lived all over the world, currently in Switzerland. Every time I have told someone that I come from Liverpool, they mention the s***e, sometimes still the Beatles (earlier it was more often, now not so much). 
99% of people couldn't even tell you where Liverpool is on a map of GB (I often have to explain to people that it is around 50 km South West of Manchester). Even less people would look at a pic of the Liver buildings and be able to tell you where it is.
What we need is for people to recognise our club instantly. That requires a unique, instantly recognisable and distinct branding. We cannot get around the fact that when football fans see the badge of the s***e, they recognise the bird as being Liverpool FC, not Liverpool the City.
In order to be instantly recognisable around the world, we need that success. An iconic stadium will help us maximise the profits to come from success on the field, a simple and uncomplicated badge will help with the ease and cost of producing merchandise but without the success, we won't be filling the stadium and won't be selling the merchandise.
Personally, I would prefer the badge to continue to feature the jail and the laurel leaves but whether we have a bird, a fish or a bloody tiger on it is irrelevant, it needs to be recognisable because people are having it shoved down their throats constantly because of our success!