Suppose $\mathbf{P} = \mathbf{BQP}$. Then what is randomness? Would it even exist at all?
P and BQP are decision-problem classes, i.e. the correct output is always a deterministic functions of the inputs. The only question is whether randomness helps "along the way" to speed up computing this deterministic function (at the cost of sometimes being wrong), or does not. This is the key point: P=BQP says nothing about outputting random strings in general situations, but only for the situation of computing a deterministic function in polynomial time.

So for instance the statement


  if P=BQP, then there exists a classical deterministic algorithm which is able to quickly output true randomness (by just quickly simulating some Quantum Turing Machine).


is false. P=BQP only implies that if a machine with randomness can usually output the correct bit, then there is a machine with no randomness that always ouptuts the correct bit.



More comments on your question:


What is closer to true is that some form of pseudorandom generators exist (deterministic machines whose output "looks random" to other deterministic machines of bounded running time). However, again, these PRGs would be targeted to machines that always output just a single bit, not all tasks that use randomness in general.
For an example single-machine problem that can be solved by a randomized machine and provably cannot be solved by a deterministic machine, consider the problem "output a string of Kolmogorov complexity $\geq n$". This is essentially the problem of outputting a random string, and of course we can prove that no deterministic machine can do this.
As Kaveh points out, in areas like communication complexity between multiple machines, also distributed systems and probably others, we have provable differences between what can be achieved with and without randomness. The classic example is for two machines to decide if they hold the same $n$-bit string while communicating fewer than $n$ bits.
It may be that even if BQP = P, the randomized algorithm is much faster and more useful in practice. (It may even be that someday complexity theory could prove this.) Example: Miller-Rabin primality test (randomized) vs AKS (deterministic).
As I understand it, we could have P = BQP and yet cryptography is perfectly useful, one-way functions exist, etc. (Not the factoring problem, of course.) And cryptography relies completely on randomness.
I think BPP is as much a model of "true" randomness as BQP. Just a different type of randomness. How to get "truly" random bits in the real world is a different story.