Argument in proving that function is not polynomial time in bit length of input seems faulty

I am currently solving a question that asks which of the following functions can be calculated in polynomial time:

$$n!, \binom{n}{5}, \binom{2n}{n}, n^{\lfloor \lg n \rfloor}, \lfloor \sqrt{n} \rfloor, \text{the smallest prime factor of } n, \text{the number of prime factors less than }n.$$

In proving the first one, I thought $$n! \geq n$$ and the input size is $$\log_2 n$$ so the output cannot even be written in polynomial time. So then clearly the calculation cannot be done in polynomial time.

But then I thought I must have some misunderstanding, since by that logic even just calculating $$n$$ from the input (that is, the identity function) should not be polynomial time. But that's clearly not possible.

What is the problem in my thinking, and instead how should I be thinking about these?

For example, when computing the identity function $$f(m) = m$$, an input $$m$$ has input length $$n = \Theta(\log m)$$ and output length also $$n$$, which is polynomial in $$n$$.
The factorial function, in contrast, has much too long output length. Indeed, if the input is $$n$$, then by Stirling's formula, the length of the output is $$\Theta(n\log n)$$, which is exponential in the input length $$\Theta(\log n)$$.