I don't think trademark protection works like that, unfortunately. I think you try to register the trademark and then anyone in opposition files their opposition to attempt to either get you to withdraw your attempt at registering it, or to get you to amend your registration (so maybe here it'd be adding FC after the word Liverpool).
I think if they do nothing though, they risk losing the right to claim a trademark on their merchandise - which means they risk losing a claim against various counterfeiters.
I think the smaller clubs will be fine using shit like City of Liverpool in their club names. I think the people who've got the most right to be annoyed are the local shirt manufacturers that make better t-shirts for LFC related stuff than the club, because they'll likely have their businesses more impact. But going back to those other clubs using "Liverpool" in their name - I think because of those clubs existing prior to the club trying to register any mark... they'll likely fail to successfully register the trademark.
But I think from a legal standpoint, if a business is seen to be doing nothing to try to protect it's intellectual property, then that can be used as evidence as to why they wouldn't care about future trademark, copyright, patent, etc... infringements in the future. I remember when there was an outcry about the attempt for the club to trademark the Liverbird (not just any Liverbird generally, but specifically the one that's got a talon over a football... the one we use on our kits). That was done over concern about counterfeiting - I suspect that this attempt to trademark the word Liverpool is the same. But I think in this instance, courts would find that you can't just trademark "Liverpool" broadly in a football sense, they'll need to add "Liverpool Football Club" or Liverpool FC" to the registration. Because there are 3 football clubs in England (that I know of) with Liverpool in the name, so I think it's unlikely they'd succeed in registering a trademark that'd fuck those clubs over.