Yeah, that makes sense. I'm working purely off memory, so while the world knew The Flash had died I am not completely clear whether they knew he was also Barry Allen initially. But the evidence does point to it becoming public knowledge between Crisis #8 and Flash #1.
There's an awful lot left unsaid about that transitional phase, presumably Barry's parents were still alive, or were they? It says something that they were never seen again after Flash #350, so If you accepted that they died shortly after Barry did then really there would be no direct relatives left alive for his secret to matter anymore. Wally as you say would logically want it made transparent to the public what happened to the original Flash and what his own relationship with him was, the better to honour Barry's memory and draw a firm line between the two.
I remember in the 1980's Mini-Series Legends, which I think predates Wally's Flash series, he was trying to figure out if he should pretend to be Barry or let the world know he was the new Flash and if he should have a secret identity or not.