Why do we assume, in dimensional analysis, that the remaining constant is dimensionless?

Walter Lewin's first lecture (at 22:16) analyzes the time $t$ for an apple to fall to the ground, using dimensional analysis. His reasoning goes like this:

It's natural to suppose that height of the apple to the ground ($h$), mass of the apple ($m$), and the acceleration due to gravity ($g$) may impact (pardon the pun) the time it takes for the apple to reach the ground. Then $$t \propto h^\alpha m^\beta g^\gamma.$$ On both sides, the units must be equivalent, so $$[T] = [L]^\alpha [M]^\beta \left[\frac{L}{T^2}\right]^\gamma = [L]^{\alpha + \gamma} [M]^\beta [T]^{-2\gamma}.$$ Therefore, $$1 = -2\gamma, \quad \alpha + \gamma = 0, \quad \beta = 0.$$ Solving, we have $$\gamma = -\frac{1}{2}, \quad \alpha = \frac{1}{2}, \quad \beta = 0.$$ Then we conclude $t = k\sqrt{\frac{h}{g}}$, where $k$ is some unit-less constant.

Lewin concludes that the apple falls independently of its mass, as proved in his thought experiment and verified in real-life. But I don't agree with his reasoning.

Lewin made the assumption that $k$ is unit-less. Why could he come to this conclusion? After all, some constants have units, like the gravitational constant ($G$).

Why isn't the following reasoning correct? The constant ($k$) has the unit $[M]^{-z}$; therefore, to match both sides of the equation, $\beta = z$. So indeed, the mass of the apple does impact its fall time.

Answer

Your example shows a fundamental idea: even though the units agree, this does not mean that the resulting equation is a law of physics. This is why physicists only 'accept' laws that have been tested experimentally.

This idea is nicely explained in the following XKCD comic:

Here, we get a more extreme example than just 'changing the units of $k$'. It turns out we could arbitrarily add different quantities to an equation, and end up with a new equation that is completely valid. This does not mean that this actually makes any sense! Your new 'law' needs to be validated with experiment, and as you can see in the comic, a single experiment may not be enough.

Dimensional analysis, then, is not used to derive new laws of physics through pure reasoning alone. Instead, your professor already knew, through whatever reason, that $t\propto h^\alpha m^\beta g^\gamma$. Even that is already a bit of a leap of faith - there is nothing⁺ that keeps you from assuming $t\propto \ln h$. To quote your own post:

[...] as proved in his thought experiment and verified in real-life.

Instead, then, you should see dimensional analysis as a very useful tool, part of a larger toolset to derive certain laws and equations. For example, if you derive an equation through whatever other means, you can use dimension analysis to check whether your new equation is possible at all. Or, in the case of this professor, you may have a general idea where your equation should be going, and you can then use dimensional analysis to get a reasonable idea of the final form of that equation. (Note: this may save you in a closed-book exam one day).

⁺That's not actually true. There are good reasons why you will probably never see $\ln h$, but that's for some other time.