If not, how do you know the assumptions are reasonable?
Yes. Starting page 58. Cobb–Douglas was derived to match empirical observations (constant factor share, constant capital and labor share) and it was proved to have some nice properties.
Note that Cobb–Douglas only approximately true starting point. It does not capture everything. For example capital and labor shares are not completely constant. The simplicity of the function makes it useful and it's taught in basic econ. classes because it holds true well enough. You need slightly more complex models or relax the function for some stuff.
Paul Douglas was a U.S. senator from Illinois from 1949 to 1966. In 1927, however, when he was still a professor of economics, he noticed a surprising fact: the division of national income between capital and labor had been roughly constant over a long period. In other words, as the economy grew more prosperous over time, the total income of workers and the total income of capital owners grew at almost exactly the same rate.
Douglas' evidence isn't actually presented in Mankiw, it's just stated as fact: do note that I asked for empirical evidence in my earlier post. The observations are over 90 years old, and the government didn't even collate national statistics in the 1920s. Do they still hold? Do they hold in non-US countries?
I'm not these modelling choices (cobb-douglas above, and the assumptions in TFA) are wrong, I'm saying that they're unjustified.