Technically, IIRC Shinra only explicitly "televised" one clip from the reused footage from the Reactor 1 raid. And both Cloud and Tifa don't appear in that clip, only Biggs, Wedge, Jessie, and briefly Barret do. So it really only stretches the suspension of disbelief in regards to Barret not being ID'ed by the public. Though Barret's conspicuousness pushing the suspension of disbelief has always been an element of the story even in the OG (though the Remake does trying to "explain" some of it by having the Slums watches exist, so it makes Slums dwellers carrying around heat a more common occurrence in Midgar, but even that only goes so far).