My wild theory is... I think Flowey went into the room and actually saw a lot of stuff relating to the brothers, but gaster was watching too and caught that Flowey was going to bring them there to remember. So gaster got rid of nearly everything so that the brothers wouldn’t have to remember the worst part of their lives.
What actually happened: Flowey saw the brothers' handplates, he explored the laboratory, saw many clues about them, put the two and two together and figured that, as a game, since he sees the world like a Player would, he'll discover something. But Gaster got erased out of existence, along with most of the proof and memories that he ever existed (Flowey would probably be able to remember him, thanks to his Determination, but he wasn't a flower when he was around Gaster back then), when he fell into the Core. So, of course that the laboratory is mostly empty.