It only shows that the Pythagorean theorem holds for that one particular set of side lengths. A mathematical proof requires that you show it is true for all possible side lengths.
An analogy would be that I take two equal-length sticks and say "given any two sticks they will be the same length". I have an example (the two sticks I'm holding) but this does not amount to a proof of my statement (and the statement is obviously incorrect).
It’s interesting to wonder if a contraption might be built whereby some slider adjusts the sizes of the two smaller squares, while maintaining the right angle (i.e. keeps the corner on a circle with the hypotenuse as diameter). It would be much more mechanically complicated and harder to build, but pretty awesome.
That's easy enough, but it changes the total area, which means some kind of drainage is required, which complicates the demonstration of equal-area squares.
As long as you don’t change the size of the large square, the total area of the two smaller squares must be the same. That’s just a restatement of the Pythagorean Theorem.
Demonstrations based on physical objects aren't mathematically precise. Who's to say that the wood doesn't flex, or that the fluid doesn't compress? You can see that the areas are equal to within measurement tolerance, but not that they're exactly equal.
Here's a classic 'physical' pictorial proof that 31.5 = 32.5:
Besides the issue that a physical demonstration only shows that the theorem is true for one particular triangle, reality is sometimes more tricky than one thinks. See for example this https://en.wikipedia.org/wiki/Missing_square_puzzle
Further, it doesn't prove exact mathematical equality. Depending on the scale of the demonstration and the accuracy of the observations, it demonstrates approximate equality up to a certain precision. (You could also demonstrate π=3.14.)
