It's not actually a lie but it's definitely misleading.
The full version is that the transfer function of each block is not fully defined without considering the source and load impedances.
The simple form of the transfer function for a single section
H(w) = 1/(1+jwRC)
is only true when fed from 0 ohms source and driving infinite load impedance.
Thus, as you correctly state, raising this to the n'th power is only correct with a buffer (Zin = inf, Zout = 0) between stages.
Allowing for the first stage as the source impedance of the second stage (and the third stage as its load impedance), the statement that the overall response is the product of each section becomes true again.
But the maths quickly becomes much more complex, hence Spice simulators...
however for some purposes, you can approximate this to some level of accuracy by decreeing R/10 to be approximately 0, and 10R to be approximately infinity, and cascading three stages with the same RC product, as R/10 * 10C, R*C, and 10R * C/10.
By minimising the loading of each stage on its predecessors, and by minimising the source impedance of each following stage, this can get close to teh desired N'th order response.
I would simulate this to find its limits, and it can't realistically be pushed beyond 2 or 3 stages.
In any case it's massively overdamped; once you introduce a buffer you are into the realms of much more optimal (e.g. Sallen and Key) filters where the second-order sections give you better control over frequency response and damping.