It only mentions new Brighton by Marine Point, so I suppose Egremont, Harrison Drive, Leasowe etc would still be okay for dog-walking. A lot of resorts have dog-free areas/times; it's not really surprising that this area wants to have a dog-free beach as part of the general upgrade, even if it is disappointing for regulars.
AS you say Greenwood, various times in other places. One big thing is that other resorts also have 'doggy bins' which for some reason is a policy that this council will not put into practice. I even saw one in Wales the other day which was in the middle of nowhere on the Mold-Denbigh Road.(Flint Council)
Never the less if we want tourists to visit the area and bring in the business, they will not come to a resort with dog messed beaches or other mess. Too much information and feedback on internet these days for people to be ignorant.
There are also new regulations being brought in by the EU too concerning the 'beach standards', and that has probably more to do with the possible changes spoken of.http://www.theguardian.com/environment/2014/may/23/blackpool-beach-eu-water-law