Why does supply equal demand?
You can easily change the informal explanation to a "quantity-based" approach, using a dynamic approach. Assume that firms in the market produce this period $S_1$ but the consumers that show up buy only $D_1&lt;S_1$. What would you do if you were a producer, for period $2$, given the information you have about where demand stands?

Assume the good is perishable. Then the unsold quantity from period $1$ is destroyed. In period $2$ are you going to produce the same amount, or something less? I would say less, to reduce somehow the quantity that will go to waste and your loss from that fact.

Assume the good is not perishable, so you have inventories left over from the first period. Now, the tendency is to produce even less than when the good is perishable, because you can sell during period $2$ the inventories from period $1$.

So we see, that starting from a quantity supplied above quantity demanded, it is reasonable to expect lower quantity supplied next period.
One can easily make the analogous argument starting with quantity demanded higher than quantity supplied. This is a gradual adjustment towards "market equilibrium", where quantity supplied will equal quantity demanded, or approximately so.

...and I didn't mention prices anywhere. You can think that this is a local market where the producers are forced to sell at a specific price, "imposed" on them by other similar markets and the expectations of consumers. In other words, what I described above can roll out while the price remains fixed from period to period.

The above was just an informal description. The benchmark approach to the dynamic adjustment towards market equilibrium is the Cobweb Theorem, see Ezekiel 1938.