That name is just a marketing scam. Unless it only serves markets within Alaska or on the border with Canada, it is not an All-Alaskan Pipeline.
What was proposed would lower the value Alaska receives for its gas as LNG is less cost-effective than pipelines for transporting Natural Gas.
Most likely the Pipeline-to-Valdez combined with LNG would serve the same lower 48 market, only via terminals and pipelines in Mexico, delivering the same gas to the same market. But the cost of transportation would be higher, the price paid in the US would be unchanged and Alaska would receive less dollar for less gas as the market would likely be less.
The Canadian route delivers Gas to an existing pipeline infrastructure that has a decline supply and surplus capacity.
Shipping longer distance to Asia and competing against Russia and Australia with lower transportation cost would result in even less return for their resouces.
But some people limit their thinking to catchy slogans.