The fact that grand pianos sound than their vertical counterparts is NOT due to the greater size of the soundboard. Just look at the examples of Yamaha pianos:
U1 (48 inch vertical), soundboard: 2137 square inches; vs.
C3 (6 foot 1 grand), soundboard: 2059.9 s/i.
(The latter costs lot more and sounds much better, but has a smaller soundboard.)
U3 (52 inch vertical), soundboard: 2369.1 s/i; vs.
C5 (6 foot 7 grand), soundboard: 2197.6 s/i.
BTW, soundboard is made of spruce not steel. Even the frame is made of cast iron, not steel.
Buyers beware: All those nonsense comes from salespeople who are too eager to make deals. They make up stories and directly pass them to you, or through your teachers, most of whom know very little about pianos. A teacher’s knowledge on pianos is as good as that of an average driver on cars. (When my auto mechanic asked me if he should ask a piano teacher to pick a piano for his wife, I asked him if he would trust me to choose a used car for him.)
More than once, I heard people here saying that the keys and the action (the moving parts inside the piano case) are the same on different sizes of the grand pianos. That is a story, not the fact. They made it up when they want to sell you a small grand. Generally speaking, a grand piano smaller than 5 footer is more a piece of furniture than a musical instrument. (Apology to 又当爹来又当妈, you got an excellent deal.)